Basic Probability

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レッスンノート

Basic Probability Concepts

Probability is a number between 0 and 1 describing likelihood: 0 = impossible, 1 = certain.
An experiment is an activity repeated to produce results; each repeat is a trial.
An outcome is a possible result of a trial; an event is an outcome or collection of outcomes.
The sample space is the set of all possible outcomes.
If all outcomes are equally likely, the probability of an event A is P(A) = (number of outcomes in A) / (total number of outcomes).
Probabilities can be given as fractions, decimals, or percentages.

Probability Scale

The probability scale runs from 0 to 1.
0 = impossible, between 0 and 0.5 = unlikely, 0.5 = even chance, between 0.5 and 1 = likely, 1 = certain.
Probabilities of all outcomes in a sample space sum to 1.
The complement of event A (not A) has probability P(not A) = 1 - P(A).

Probability scale

Mutually Exclusive Events

Two events are mutually exclusive if they cannot happen at the same time.
For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
Complementary events are always mutually exclusive.

Calculating Basic Probabilities

If all outcomes equally likely, probability of a specific outcome = 1 / total number of outcomes.
Example: A bag with 50 marbles, 20 blue → P(blue) = 20/50 = 2/5.
To find a missing probability in a table, make all probabilities sum to 1.
If two probabilities are equal and their sum is known, divide by 2 to find each.

Possibility (Sample Space) Diagrams

A possibility diagram (or sample space diagram) is a grid showing all outcomes when combining two sets of outcomes.
For example, rolling two dice: grid shows sums from 2 to 12, with 36 equally likely outcomes.
Probabilities are found by counting favourable outcomes and dividing by total outcomes.
This method only works if all outcomes in the sample space are equally likely.

Sample space for two dice

Relative Frequency

Relative frequency = (number of successful trials) / (total number of trials).
It is an estimate of probability based on experiment, used when theoretical probability is unknown.
The more trials, the more accurate the estimate becomes.
Relative frequency can be compared to theoretical probability to test fairness or bias.

Expected Frequency

Expected frequency = probability × number of trials.
It gives the number of times you would expect an outcome to occur in a given number of trials.
Example: If P(yellow) = 1/3 and 300 trials, expected yellow = 300 × 1/3 = 100.
Expected frequency can be based on theoretical probability or relative frequency.

Key Terminology

Fair means all outcomes equally likely; biased means not equally likely.
n(A) denotes the number of outcomes in event A.
P(A) denotes the probability of event A.
Trials must be independent and random for relative frequency to be valid.

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練習問題

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1.A fair spinner has the numbers 2, 2, 3, 4, 4, 5. What is the probability that it lands on an even number?
Easy
- A1/2
- B2/3
- C1/3
- D5/6
2.The probability that it rains tomorrow is 0.35. What is the probability that it does not rain tomorrow?
Easy
- A0.65
- B0.35
- C0.5
- D0.75
3.A box contains 22 pencils: 6 pink, 9 blue, 7 yellow. What is the probability of taking a green pencil?
Easy
- A0
- B1/22
- C7/22
- D6/22
4.The probability that Alex wins a prize is 0.27. What is the probability that Alex does not win a prize?
Easy
- A0.73
- B0.27
- C0.5
- D0.63
5.A bag contains 6 red and 10 blue balls. What is the probability of taking a blue ball?
Easy
- A10/16
- B6/16
- C10/10
- D1/2
6.A bag contains 20 balls, 5 of which are red. A ball is picked at random. On a probability scale from 0 to 1, which arrow shows the probability that the ball is red?
Easy
- AA
- BB
- CC
- DD
7.A spinner has 8 sides numbered 3, 4, 4, 7, 7, 7, 8, 9. What is the probability of landing on 7?
Easy
- A3/8
- B1/8
- C1/3
- D3/7
8.The probability of getting a seat on the bus is 0.93. There are 200 school days. What is the expected number of days Marcus will not get a seat?
Medium
- A14
- B186
- C7
- D93

もっと問題

9.A bag contains 3 blue, 4 red, 8 green pens. A pen is chosen at random. What is the probability it is blue?
Easy
- A3/15
- B4/15
- C8/15
- D1/3
10.A bag contains counters: red, blue, green, yellow. Probabilities: red 0.32, blue 0.2, yellow 0.16. What is the probability of green?
Medium
- A0.32
- B0.28
- C0.16
- D0.2
11.A bag contains 7 red, 5 green, 2 pink discs. Helen picks a disc at random, records colour, replaces it. She does this 140 times. How many times would she expect a green disc?
Medium
- A50
- B35
- C70
- D14
12.Two fair six-sided dice are rolled. What is the probability that the sum is an odd number greater than 5?
Hard
- A1/3
- B5/18
- C1/2
- D7/36
13.Two fair spinners: spinner A numbers 1,2,2,4; spinner B numbers 3,3,4,5. Both are spun and the numbers added. What is the probability the total is 5?
Medium
- A3/16
- B1/4
- C5/16
- D1/8
14.A bag contains 6 green, 5 red, 3 blue balls. A ball is taken at random. What is the probability it is green or red?
Easy
- A11/14
- B6/14
- C5/14
- D1/2
15.A bag contains brown, white, black, purple balls only. Probabilities: brown 0.46, white 0.22, black 0.14. There are 50 balls. How many black balls are there?
Medium
- A7
- B9
- C11
- D14
16.A bag contains 3 green balls, 4 red balls and 1 blue ball only. Matt takes a ball from the bag at random. What is the probability that he takes a red ball?
Easy
- A4/8
- B3/8
- C1/8
- D4/7
17.A bag contains 20 balls, 5 of which are red. A ball is picked at random. What is the probability that it is red?
Easy
- A5/20
- B15/20
- C5/15
- D20/5
18.A fair five-sided spinner is numbered 2, 3, 4, 5, 6. What is the probability that it lands on an odd number?
Easy
- A2/5
- B3/5
- C1/5
- D4/5
19.A bag contains 6 blue, 4 red and 5 yellow counters. One counter is drawn at random and then replaced. What is the probability that the counter is yellow?
Easy
- A1/3
- B5/11
- C5/10
- D1/5
20.The probability that it will be sunny tomorrow is 0.97. What is the probability that it will not be sunny tomorrow?
Easy
- A0.03
- B0.97
- C0.3
- D0.07
21.The probability that a sweet made in a factory is the wrong shape is 0.0028. One day, the factory makes 25000 sweets. Calculate the number of sweets expected to be the wrong shape.
Easy
- A70
- B28
- C700
- D280
22.A bag contains 7 red, 5 green, 2 pink discs. Helen picks a disc at random, records colour, replaces it. She does this 140 times. How many times would she expect to take a pink disc?
Medium
- A20
- B10
- C35
- D28
23.A fair 8-sided spinner has numbers 3, 4, 4, 7, 7, 7, 8, 9. The spinner is spun 160 times. How many times would you expect it to land on 7?
Medium
- A60
- B40
- C80
- D20
24.A fair six-sided dice is rolled. What is the probability of rolling a prime number?
Easy
- A1/2
- B2/3
- C5/6
- D1/3
25.A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. One marble is taken at random. What is the probability that it is blue?
Easy
- A0.3
- B0.5
- C0.2
- D0.1
26.The probability that a train is late is 0.15. What is the probability that it is not late?
Easy
- A0.85
- B0.15
- C0.75
- D0.95
27.A bag contains only red, blue, and green counters. The probability of picking a red counter is 0.4 and the probability of picking a blue counter is 0.25. What is the probability of picking a green counter?
Easy
- A0.35
- B0.65
- C0.45
- D0.55
28.A fair spinner has 8 equal sections numbered 1 to 8. It is spun 200 times. How many times would you expect it to land on a number greater than 5?
Medium
- A75
- B100
- C50
- D125
29.Two fair six-sided dice are rolled. What is the probability that the sum of the two dice is 7?
Medium
- A1/6
- B1/12
- C5/36
- D1/9
30.Which of the following is the correct notation for the probability of event A?
Easy
- AP(A)
- Bn(A)
- CA(P)
31.The probability of an event can be greater than 1.
Easy
True or false?
32.A bag contains only red and blue balls. The probability of picking a red ball is 0.35. What is the probability of picking a blue ball?
Easy
- A0.65
- B0.35
- C0.5
- D0.7
33.The probabilities of a spinner landing on red, blue, and green are 0.2, 0.3, and 0.1 respectively. What is the probability of landing on yellow?
Medium
- A0.4
- B0.5
- C0.6
- D0.3
34.Mutually exclusive events can occur at the same time.
Easy
True or false?
35.Which of the following are possible values for a probability? (Select all that apply)
Easy
- A0
- B0.5
- C1
- D1.5
- E-0.2
36.The probability of an event A is 0.45. What is the probability of the complement of A?
Easy
- A0.55
- B0.45
- C0.5
- D0.65
37.Match each term to its correct definition.
Medium
- Probability
- Relative frequency
- Expected frequency
- Sample space
- A number between 0 and 1 describing likelihood
- Estimate of probability from an experiment
- Number of times an outcome is expected to occur
- Set of all possible outcomes
38.Place these probability values in order from impossible to certain.
Easy
- 0.5
- 0
- 1
- 0.2
39.Which of the following statements about probability are true? (Select all that apply)
Medium
- AThe probability of an event can be greater than 1.
- BThe sum of probabilities of all possible outcomes is 1.
- CMutually exclusive events can occur at the same time.
- DThe complement of an event has probability 1 minus the event's probability.
- ERelative frequency becomes more accurate with more trials.
40.A fair six-sided dice is rolled. What is the probability of rolling a number less than 3? Give your answer as a fraction in simplest form.
Easy
41.Complete the sentence.
Easy
The probability scale ranges from ____ to ____.
42.What is the term for an event that has no chance of occurring?
Easy
43.A fair coin is flipped once. What is the probability of getting tails? Give your answer as a decimal.
Easy
44.Explain what is meant by 'mutually exclusive events' and give an example.
Medium
45.A fair six-sided dice is rolled. What is the probability of rolling a number less than 3? Give your answer as a fraction in simplest form.
Easy
- A1/3
- B1/2
- C1/6
- D2/3
46.A fair coin is flipped once. What is the probability of getting tails? Give your answer as a decimal.
Easy
- A-0.5
- B0.5
- C1.5
- D2.5

復習ノート

## Basic Probability - **Probability scale**: 0 (impossible) to 1 (certain). - **Key formula**: \( P(A) = \frac{\text{number of outcomes for event } A}{\text{total number of outcomes}} \) (equally likely outcomes). - **Complement**: \( P(\text{not } A) = 1 - P(A) \). - **Mutually exclusive events**: \( P(A \text{ or } B) = P(A) + P(B) \). - **All probabilities sum to 1**. ## Possibility (Sample Space) Diagrams - List or grid showing all possible outcomes. - Count desired outcomes and divide by total. ## Relative & Expected Frequency - **Relative frequency**: \( \frac{\text{number of successful trials}}{\text{total number of trials}} \). - **Expected frequency**: \( \text{probability} \times \text{number of trials} \). - More trials → better estimate. **Tips**: Use fractions unless told otherwise. For "not" questions, use complement. For "or" with mutually exclusive, add probabilities.

Probability: A number between 0 and 1 that describes the likelihood of an event occurring.
Sample space: The set of all possible outcomes of an experiment.
Event: An outcome or a collection of outcomes.
Mutually exclusive: Events that cannot happen at the same time.
Complement: The event that the original event does not occur; P(not A) = 1 - P(A).
Relative frequency: An estimate of probability based on experimental data: number of successful trials divided by total trials.
Expected frequency: The number of times an outcome is expected to occur: probability × number of trials.

フラッシュカード

What is the probability scale?

A scale from 0 to 1 where 0 means impossible, 1 means certain, 0.5 means even chance.

How do you calculate basic probability?

P(A) = (number of outcomes for event A) / (total number of equally likely outcomes).

What is the complement rule?

P(event does not happen) = 1 - P(event happens).

What are mutually exclusive events?

Events that cannot happen at the same time. For mutually exclusive events, P(A or B) = P(A) + P(B).

What is a sample space?

The set of all possible outcomes of an experiment. It can be listed or shown in a table.

What is a possibility diagram?

A grid showing all possible outcomes when combining two sets of outcomes, e.g., rolling two dice.

How do you find probability from a possibility diagram?

Count the number of desired outcomes and divide by the total number of outcomes in the diagram.

What is relative frequency?

An estimate of probability based on experiment: (number of successful trials) / (total number of trials).

What is expected frequency?

The number of times you expect an outcome to occur: probability × number of trials.

A fair spinner has numbers 2,2,3,4,4,5. Find P(even number).

There are 4 even numbers (2,2,4,4) out of 6. P(even) = 4/6 = 2/3.

A bag has 1,1,2,3,3,3,4,5. Find P(odd number).

Odd numbers: 1,1,3,3,3,5 (6 odds) out of 8. P(odd) = 6/8 = 3/4.

A five-sided spinner numbered 2,3,4,5,6. Find P(prime number).

Prime numbers: 2,3,5 (3 primes) out of 5. P(prime) = 3/5.

30 flags: 11 pink, 7 yellow, 5 blue, 4 silver, 3 green. Find P(not blue).

Not blue = total - blue = 30 - 5 = 25. P(not blue) = 25/30 = 5/6.

A bag has 3 green, 4 red, 1 blue. Find P(red).

4 red out of 8 total. P(red) = 4/8 = 1/2.

A bag has 6 red and 10 blue balls. Find P(blue).

10 blue out of 16 total. P(blue) = 10/16 = 5/8.

P(green) = 0.38. Find P(red) if only green and red.

P(red) = 1 - 0.38 = 0.62.

P(win) = 0.27. Find P(not win).

P(not win) = 1 - 0.27 = 0.73.

P(sunny) = 0.97. Find P(not sunny).

P(not sunny) = 1 - 0.97 = 0.03.

P(rain) = 0.35. Find P(not rain).

P(not rain) = 1 - 0.35 = 0.65.

A bag has 20 balls, 5 red. Find P(red).

5 red out of 20. P(red) = 5/20 = 1/4.

P(cycle) = 1/5. Find P(walk) if only walk or cycle.

P(walk) = 1 - 1/5 = 4/5.

Expected number of days Dan cycles in 200 days if P(cycle)=1/5.

Expected = 200 × 1/5 = 40 days.

A spinner has numbers 3,4,4,7,7,7,8,9. Find P(7).

Three 7s out of 8. P(7) = 3/8.

Spinner from previous card spun 160 times. Expected number of 7s.

Expected = 160 × 3/8 = 60.

P(wrong shape)=0.0028, 25000 sweets made. Expected wrong shape?

Expected = 25000 × 0.0028 = 70.

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