Probability scale and single events

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课程笔记

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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练习题

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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

复习笔记

## 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.

闪卡

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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