Probability scale and single events

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The Language of Probability

Probability describes how likely an event is to happen, using words such as impossible, unlikely, even chance, likely and certain.
An impossible event cannot happen, for example rolling a 7 on an ordinary six-sided dice.
An even chance means an event is just as likely to happen as not to happen, such as getting heads when tossing a fair coin.
A certain event will definitely happen, for example getting a number less than 7 when rolling an ordinary dice.
Words like unlikely and likely sit between impossible and certain, but they do not give an exact value.

The Probability Scale

Probabilities are measured on a scale from 0 to 1.
A probability of 0 means the event is impossible.
A probability of 1 means the event is certain.
A probability of 0.5 (or 1/2) means an even chance.
The scale can also be written as percentages from 0% to 100%.
The larger the probability, the more likely the event is to occur.

Equally Likely Outcomes

When all outcomes are equally likely, each outcome has the same chance of happening.
A fair coin has two equally likely outcomes: heads and tails.
A fair six-sided dice has six equally likely outcomes: 1, 2, 3, 4, 5 and 6.
Tossing a coin twice gives four equally likely outcomes: head-head, head-tail, tail-head and tail-tail.

Calculating Probability of a Single Event

For equally likely outcomes, the probability of an event is: P = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}.
The answer can be written as a fraction, a decimal or a percentage.
Example: the probability of getting heads on a fair coin is 1/2 = 0.5 = 50%.
Example: the probability of rolling a 4 on a fair dice is 1/6 ≈ 0.167 ≈ 16.7%.
Example: tossing two coins, the probability of head-head is 1/4 = 0.25 = 25%.
Example: tossing two coins, the probability of at least one head is 3/4 = 0.75 = 75%.

Probabilities of All Outcomes Sum to 1

The probabilities of all possible outcomes of an event add up to 1 (or 100%).
For a fair coin: P(heads) + P(tails) = 1/2 + 1/2 = 1.
For a fair dice: P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1.
This rule helps you check that you have not missed any outcomes.

Probability of an Event Not Happening

If the probability of an event happening is P, the probability of it not happening is 1 − P.
Example: if P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7.
Example: if P(winning a game) = 1/5, then P(not winning) = 1 − 1/5 = 4/5.
The probability of an event and the probability of its complement always add up to 1.

Theoretical vs Empirical Probability

Theoretical probability is found by counting equally likely outcomes, as in tossing a fair coin.
Empirical probability (or experimental probability) is found by carrying out an experiment and recording results.
Theoretical probability is based on reasoning; empirical probability is based on data.
In the long run, empirical probability tends to get closer to theoretical probability.

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練習問題

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1.Which of these numbers cannot be the probability of an event?
Easy
- A0
- B0.5
- C1.2
- D1
2.The probability that it will rain tomorrow is 0.35. What is the probability, as a decimal, that it will not rain tomorrow?
Easy
- A0.35
- B0.55
- C0.65
- D0.75
3.A fair coin is tossed once. Which list gives the probability of heads written as a fraction, a decimal and a percentage?
Easy
- A1/2, 0.5, 50%
- B1/2, 0.2, 20%
- C1/4, 0.25, 25%
- D1/1, 1.0, 100%
4.Which of the following are valid ways to write the probability of an event that is certain to happen? (select all that apply)
Medium
- A0
- B1
- C100%
- D1/2
- E0.1
5.The probability of an event not happening is 1 minus the probability of the event happening.
Easy
True or false?
6.If the probability of an event is 0.8, the event is impossible.
Easy
True or false?
7.Match each probability to the best description of how likely the event is.
Medium
- 0
- 0.5
- 1
- certain
- even chance
- impossible
8.Put these probabilities in order from least likely to most likely.
Medium
- 0.1
- 0.5
- 0.75
- 1

もっと問題

9.Match each probability to the correct description of how likely the event is.
Easy
- 0
- 0.25
- 0.5
- 1
- impossible
- unlikely
- even chance
- certain
10.Which of the following could be the probability of an event? (select all that apply)
Medium
- A0
- B1
- C0.5
- D1.5
- E-0.2
11.The probability of an event that is certain to happen is 1.
Easy
True or false?
12.A fair coin is tossed once. What is the probability of getting tails?
Easy
- A0
- B0.5
- C1
- D0.25
13.A bag contains 5 red counters and 15 blue counters. A counter is taken at random. What is the probability that it is red? Give your answer as a fraction in its simplest form.
Easy
- A\frac{1}{3}
- B\frac{1}{4}
- C\frac{1}{5}
- D\frac{1}{20}
14.A fair six-sided dice is rolled once. What is the probability that it lands on a prime number? Give your answer as a fraction.
Medium
- A\frac{1}{6}
- B\frac{1}{3}
- C\frac{1}{2}
- D\frac{2}{3}
15.A fair coin is tossed once. Which of the following statements are true? (select all that apply)
Medium
- AThe probability of heads is 0.5.
- BThe probability of tails is 50%.
- CThe probability of heads is greater than the probability of tails.
- DThe probability of heads is \frac{1}{2}.
- EThe probability of tails is 1.
16.A fair coin is tossed once. What is the probability of getting tails? Give your answer as a fraction.
Easy
- A\frac{1}{4}
- B\frac{1}{2}
- C\frac{3}{4}
- D1
17.A fair six-sided dice is rolled once. What is the probability that it lands on a number less than 5? Give your answer as a decimal.
Medium
- A0.3
- B0.5
- C0.67
- D0.83
18.A bag contains 3 red, 4 blue and 5 green counters. A counter is taken at random. What is the probability that it is not green? Give your answer as a fraction in its simplest form.
Medium
- A\frac{5}{12}
- B\frac{7}{12}
- C\frac{1}{2}
- D\frac{3}{4}
19.A spinner has 8 equal sections numbered 1 to 8. What is the probability that the spinner lands on a multiple of 3? Give your answer as a percentage.
Medium
- A12.5%
- B25%
- C37.5%
- D50%
20.A fair coin is tossed twice. What is the probability of getting two heads? Give your answer as a fraction.
Medium
- A\frac{1}{4}
- B\frac{1}{2}
- C\frac{3}{4}
- D1
21.A fair six-sided dice is rolled once. What is the probability that it lands on a number that is a factor of 6? Give your answer as a fraction.
Medium
- A\frac{1}{6}
- B\frac{1}{3}
- C\frac{1}{2}
- D\frac{2}{3}
22.A fair 10-sided dice, numbered 1 to 10, is rolled once. What is the probability that it lands on an even number? Give your answer as a percentage.
Medium
- A10%
- B20%
- C50%
- D80%
23.A bag contains 4 red, 3 blue and 5 yellow counters. A counter is taken at random. What is the probability that it is blue? Give your answer as a fraction in its simplest form.
Medium
- A\frac{1}{4}
- B\frac{1}{3}
- C\frac{5}{12}
- D\frac{1}{2}
24.A fair six-sided dice is rolled once. What is the probability that it lands on a 7? Give your answer as a fraction.
Medium
- A0
- B\frac{1}{6}
- C\frac{1}{2}
- D1
25.A fair six-sided dice is rolled once, and a fair coin is tossed once. What is the probability of getting an even number on the dice and heads on the coin? Give your answer as a fraction.
Medium
- A\frac{1}{4}
- B\frac{1}{2}
- C\frac{1}{3}
- D\frac{2}{3}
26.A spinner has 10 equal sections numbered 1 to 10. What is the probability that the spinner lands on a number that is both even and greater than 6? Give your answer as a fraction in its simplest form.
Medium
- A\frac{1}{5}
- B\frac{3}{10}
- C\frac{2}{5}
- D\frac{1}{2}
27.A bag contains 6 red counters and 4 blue counters. A counter is taken at random and not replaced. A second counter is then taken at random. What is the probability that both counters are red? Give your answer as a fraction in its simplest form.
Medium
- A\frac{1}{3}
- B\frac{9}{25}
- C\frac{3}{5}
- D\frac{2}{5}
28.A fair coin is tossed three times. What is the probability of getting at least one head? Give your answer as a fraction.
Hard
- A\frac{7}{8}
- B\frac{3}{8}
- C\frac{1}{8}
- D\frac{1}{2}
29.A fair six-sided dice is rolled once and a fair coin is tossed once. What is the probability of getting a 6 on the dice or heads on the coin?
Medium
- A\frac{7}{12}
- B\frac{2}{3}
- C\frac{1}{12}
- D\frac{1}{2}
30.A fair 8-sided dice, numbered 1 to 8, is rolled once. What is the probability that it lands on a factor of 12? Give your answer as a fraction in its simplest form.
Medium
- A\frac{3}{8}
- B\frac{1}{2}
- C\frac{1}{4}
- D\frac{5}{8}
31.The probability that a biased dice lands on a 6 is 0.2. The dice is rolled 50 times. What is the best estimate for the number of times it lands on a 6?
Medium
- A10
- B5
- C20
- D25
32.A fair coin is tossed twice. The outcomes are HH, HT, TH and TT, each equally likely. What is the probability of getting exactly one head? Give your answer as a fraction.
Medium
- A\frac{1}{2}
- B\frac{1}{4}
- C\frac{3}{4}
- D\frac{1}{3}
33.A bag contains 5 red, 3 blue and 2 green counters. One counter is taken at random. What is the probability that it is red or blue? Give your answer as a fraction in its simplest form.
Medium
- A\frac{4}{5}
- B\frac{1}{2}
- C\frac{3}{10}
- D\frac{7}{10}
34.A fair six-sided dice is rolled twice. What is the probability that the total score is 12? Give your answer as a fraction.
Hard
- A\frac{1}{36}
- B\frac{1}{6}
- C\frac{1}{12}
- D\frac{1}{18}
35.A fair 10-sided dice, numbered 1 to 10, is rolled once. What is the probability that it lands on a number that is a multiple of 4? Give your answer as a decimal.
Medium
- A0.2
- B0.4
- C0.25
- D0.5
36.A fair coin is tossed four times. What is the probability of getting at least one tail? Give your answer as a fraction.
Medium
- A\frac{15}{16}
- B\frac{1}{16}
- C\frac{1}{2}
- D\frac{7}{8}
37.On the probability scale, what is the lowest possible value?
Easy
- A0
- B1
- C0.5
- D-1
38.Which of these is the correct way to write the probability of a fair coin landing on heads?
Easy
- A\frac{1}{2}
- B\frac{2}{1}
- C2
- D0.2
39.The probability of any event is a number between 0 and 1 inclusive.
Easy
True or false?
40.Which of the following are valid ways to express the probability that a fair coin lands on heads? (select all that apply)
Medium
- A\frac{1}{2}
- B0.5
- C50%
- D2
- E1.5
41.What word describes an event with a probability of 0.5?
Easy
42.A fair six-sided dice is rolled once. What is the probability that it lands on a 4? Give your answer as a fraction.
Easy
43.Complete the sentence about the probability scale.
Easy
A probability of ____ means the event is impossible, and a probability of ____ means the event is certain.
44.What word describes an event with a probability of 0?
Easy
45.A fair six-sided dice is rolled once. What is the probability that it lands on a number greater than 4? Give your answer as a decimal.
Medium
46.A fair 10-sided dice, numbered 1 to 10, is rolled once. What is the probability that it lands on a number greater than 7? Give your answer as a decimal.
Medium
47.A fair coin is tossed once. What is the probability of getting heads? Give your answer as a decimal.
Medium
48.What word describes an event with a probability of 1?
Medium
49.On the probability scale, what word describes an event that is just as likely to happen as not to happen?
Easy
50.A fair coin is tossed once. Give the probability of getting heads as a decimal.
Easy
51.Complete the sentences about the probability scale using the words from the list.
Easy
An event with probability 0 is ____. An event with probability 1 is ____. An event with probability 0.5 has an ____ chance.
52.A fair six-sided dice is rolled once. What is the probability that it lands on an odd number? Give your answer as a fraction.
Medium
53.A fair coin is tossed twice. Explain how to find the probability of getting at least one head, and give the probability.
Medium
54.Explain how to find the probability of getting at least one head when a fair coin is tossed twice, and give the probability as a fraction, decimal and percentage.
Hard
55.A fair coin is tossed twice. Explain how to find the probability of getting at least one head, and give the probability as a fraction, decimal and percentage.
Hard
56.A fair six-sided dice is rolled once. Explain how to find the probability that it lands on a prime number, and state the probability as a fraction in its simplest form.
Hard
57.Explain what is meant by an even chance, and give an example with its probability.
Medium
58.A fair six-sided dice is rolled once. What is the probability that it lands on a 4? Give your answer as a fraction.
Easy
- A0.16666666666666666
- B-0.8333
- C2.1667
- D1.1667
59.A fair six-sided dice is rolled once. What is the probability that it lands on a number greater than 4? Give your answer as a decimal.
Medium
- A2.3333
- B1.3333
- C-0.6667
- D0.3333333333333333
60.A fair 10-sided dice, numbered 1 to 10, is rolled once. What is the probability that it lands on a number greater than 7? Give your answer as a decimal.
Medium
- A-0.7
- B2.3
- C0.3
- D1.3
61.A fair coin is tossed once. What is the probability of getting heads? Give your answer as a decimal.
Medium
- A2.5
- B0.5
- C1.5
- D-0.5
62.A fair coin is tossed once. Give the probability of getting heads as a decimal.
Easy
- A2.5
- B0.5
- C-0.5
- D1.5
63.A fair six-sided dice is rolled once. What is the probability that it lands on an odd number? Give your answer as a fraction.
Medium
- A2.5
- B1.5
- C-0.5
- D0.5

復習ノート

## Probability Scale and Single Events **Probability** measures how likely an event is to happen. It is always a number between 0 and 1, and can be written as a fraction, decimal or percentage (0% to 100%). ### The probability scale - **0** = impossible (cannot happen) - **0.5** = even chance (equally likely to happen or not) - **1** = certain (must happen) Words used along the scale: impossible, unlikely, even chance, likely, certain. ### Equally likely outcomes A fair coin has two equally likely outcomes: heads and tails. So: \(P(\text{heads}) = \frac{1}{2} = 0.5 = 50\%\) (P = probability.) ### Calculating the probability of a single event When all outcomes are equally likely: \(P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}\) **Example:** tossing a fair coin twice gives four equally likely outcomes: head-head, head-tail, tail-head, tail-tail. - P(head-head) = 1/4 = 0.25 = 25% - P(at least one head) = 3/4 = 0.75 = 75% ### All outcomes sum to 1 The probabilities of all possible outcomes add up to 1. For a fair coin: 1/2 + 1/2 = 1. ### Probability of an event not happening If the probability an event happens is P, then the probability it does **not** happen is: \(P(\text{not } E) = 1 - P(E)\) **Example:** if P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7. ### Quick checks - A probability can never be negative or greater than 1. - A larger probability means the event is more likely. - Theoretical probability uses counting outcomes; empirical probability comes from real experiments.

Probability: A number between 0 and 1 describing how likely an event is to occur; larger values mean more likely.
Probability scale: The range from 0 to 1 (or 0% to 100%) on which probabilities are placed, from impossible to certain.
Impossible: An event that cannot happen; its probability is 0.
Certain: An event that must happen; its probability is 1.
Even chance: An event that is equally likely to happen or not; its probability is 0.5 (50%).
Equally likely outcomes: Outcomes that all have the same chance of occurring, such as heads and tails on a fair coin.
Single event: One outcome or set of outcomes whose probability is being found.
Theoretical probability: Probability found by dividing the number of favourable outcomes by the total number of possible outcomes.
Empirical probability: Probability based on the results of real experiments rather than counting outcomes.
Complement: The event that a given event does not happen; its probability is 1 minus the probability of the event.

フラッシュカード

What is the probability scale?

A scale from 0 to 1 (or 0% to 100%) that measures how likely an event is to occur. 0 means impossible and 1 means certain.

What does a probability of 0 mean?

The event is impossible; it will never happen.

What does a probability of 1 mean?

The event is certain; it will definitely happen.

What does a probability of 0.5 (or 1/2, 50%) mean?

The event has an even chance of happening; it is equally likely to happen as not.

Describe the likelihood of an event with probability 0.1.

It is unlikely to happen, because 0.1 is close to 0.

Describe the likelihood of an event with probability 0.9.

It is likely to happen, because 0.9 is close to 1.

What are equally likely outcomes?

Outcomes that have the same chance of occurring, such as heads and tails when tossing a fair coin.

How do you calculate the probability of a single event when all outcomes are equally likely?

P(event) = (number of favourable outcomes) / (total number of possible outcomes).

What is the probability of rolling a 3 on a fair six-sided die?

1/6, because there is 1 favourable outcome (rolling a 3) out of 6 equally likely outcomes.

What is the probability of getting heads when tossing a fair coin?

1/2, 0.5 or 50%, because there are 2 equally likely outcomes and 1 is heads.

If you toss a fair coin twice, what is the probability of getting two heads?

1/4, 0.25 or 25%, because there are 4 equally likely outcomes (HH, HT, TH, TT) and only 1 is two heads.

If you toss a fair coin twice, what is the probability of getting at least one head?

3/4, 0.75 or 75%, because 3 of the 4 equally likely outcomes (HH, HT, TH) contain at least one head.

What is the sum of the probabilities of all possible outcomes of an event?

The sum is always 1 (or 100%).

How do you find the probability of an event not happening?

P(not A) = 1 − P(A), where P(A) is the probability of the event happening.

If the probability of rain tomorrow is 0.3, what is the probability that it does not rain?

0.7, because 1 − 0.3 = 0.7.

What is the probability of rolling a number greater than 4 on a fair six-sided die?

1/3, because the favourable outcomes are 5 and 6 (2 outcomes) out of 6, so 2/6 = 1/3.

What is the probability of rolling an even number on a fair six-sided die?

1/2, because the favourable outcomes are 2, 4 and 6 (3 outcomes) out of 6, so 3/6 = 1/2.

What is the probability of drawing a red card from a standard deck of 52 playing cards?

1/2, because there are 26 red cards out of 52, so 26/52 = 1/2.

What is the probability of drawing an ace from a standard deck of 52 playing cards?

1/13, because there are 4 aces out of 52, so 4/52 = 1/13.

What is the probability of selecting a vowel at random from the letters A, B, C, D, E?

2/5, because there are 2 vowels (A and E) out of 5 letters.

What is the probability of selecting a consonant at random from the letters A, B, C, D, E?

3/5, because there are 3 consonants (B, C, D) out of 5 letters.

What is the probability of rolling a 7 on a fair six-sided die?

0, because it is impossible to roll a 7 on a standard six-sided die.

What is the probability of rolling a number less than 7 on a fair six-sided die?

1, because it is certain that any roll will be less than 7.

What is the probability of drawing a blue marble from a bag containing only red marbles?

0, because there are no blue marbles, so the event is impossible.

What is the probability of drawing a red marble from a bag containing only red marbles?

1, because every marble is red, so the event is certain.

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