Conditional Probability

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수업 노트

What is Conditional Probability?

A conditional probability is the probability of an event A occurring given that event B has already occurred.
It is written as P(A | B), meaning 'probability of A given B'.
The probability is calculated from a restricted set of outcomes (only those where B has happened), not from all possible outcomes.
For example, picking a multiple of 3 from even numbers {2,4,6,8} gives P(multiple of 3 | even) = 1/4.

Calculating Conditional Probabilities from Tables, Venn Diagrams, and Tree Diagrams

Conditional probabilities are often found using two-way tables, Venn diagrams, or tree diagrams.
From a Venn diagram: P(A | B) = (number in A∩B) / (number in B).
From a two-way table: restrict to the row/column of the given condition, then find the required fraction.
Tree diagrams naturally show conditional probabilities along branches (e.g., second draw probabilities change after first draw without replacement).

P(A | B) from a Venn diagram

Combined Conditional Probabilities (Without Replacement)

When drawing two or more items without replacement, the probability of the second event depends on the outcome of the first.
Multiply probabilities along branches: P(A then B) = P(A) × P(B | A).
For example, drawing two red marbles from a bag of 4 red and 2 yellow: P(RR) = (4/6) × (3/5) = 12/30 = 2/5.
Always adjust the total number of items and the number of favourable items after each draw.

4 red, 2 yellow — without replacement

Using the 'Or' Rule for Multiple Cases

When there are multiple ways to achieve a result (e.g., different colour orders), calculate each case separately and add the probabilities.
For two items of different colours from a bag of 10 yellow, 6 blue, 4 green: P(different) = P(YB) + P(BY) + P(YG) + P(GY) + P(BG) + P(GB).
Alternatively, use 1 − P(same colour) = 1 − [P(YY)+P(BB)+P(GG)].
Remember that order matters: YB is different from BY.

10 yellow, 6 blue, 4 green — without replacement

Working with Algebraic Probabilities

When the number of items is unknown, use algebra to set up equations.
Let the unknown number be a variable (e.g., x blue marbles, x+3 red marbles).
Write the probability of the required outcome in terms of x and set it equal to the given probability.
Solve the resulting equation (often a quadratic) to find the value of x.

Common Pitfalls and Tips

Do not simplify fractions too early; keep a common denominator for easy addition.
Read carefully: 'given that' signals a conditional probability.
For 'at least one' problems, it is often easier to use 1 − P(none).
In tree diagrams, label branches clearly and check that probabilities on branches from the same node sum to 1.

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연습 문제

무료 미리 보기 — 43개 중 8개 문제. 가입하면 전부 볼 수 있어요.

1.A bag contains 4 red marbles and 2 yellow marbles. Behnaz picks two marbles at random without replacement. What is the probability that both marbles are red?
Easy
- A2/5
- B1/3
- C4/15
- D2/15
2.A bag contains 5 blue marbles and 2 green marbles. Bryn picks one marble at random without replacement. If this marble is not green, he picks another marble at random without replacement. He continues until he picks a green marble. What is the probability that he picks a green marble on his first, second or third attempt?
Medium
- A5/7
- B20/21
- C1/21
- D2/7
3.Bag A contains 3 black balls and 2 white balls. Bag B contains 1 black ball and 3 white balls. A ball is taken at random from each bag. What is the probability that the two balls have different colours?
Medium
- A9/20
- B11/20
- C1/2
- D3/10
4.A group of 200 people were asked which city they would like to visit next. The table shows the results: | City | Frequency | |---|---| | London | 50 | | Paris | 48 | | New York | 56 | | Tokyo | 46 | Two people are chosen at random from the group of 200. What is the probability that one person would like to visit London next and the other person would like to visit New York next? Give your answer as a percentage.
Medium
- A14%
- B28%
- C7%
- D56%
5.The speeds of 200 cars are measured. The table shows frequencies: | Speed, v (km/h) | Frequency | |---|---| | 0 < v ≤ 20 | 16 | | 20 < v ≤ 40 | 34 | | 40 < v ≤ 45 | 62 | | 45 < v ≤ 50 | 58 | | 50 < v ≤ 60 | 26 | | 60 < v ≤ 80 | 4 | Two of the 200 cars are chosen at random. What is the probability that they both have a speed greater than 50 km/h?
Medium
- A30/200 × 29/199
- B30/200 × 30/200
- C30/200 × 29/200
- D30/200 × 30/199
6.Ravi has a bag which contains 10 red balls and 8 blue balls. Ravi takes two balls at random from his bag, without replacement. What is the probability that one ball is red and one ball is blue?
Medium
- A80/153
- B10/18 × 8/17
- C10/18 × 8/17 × 2
- D10/18 × 8/17 + 8/18 × 10/17
7.A box contains 20 packets of potato chips. 6 packets contain barbecue flavour, 10 packets contain salt flavour, 4 packets contain chicken flavour. Maria takes two packets at random without replacement. What is the probability that she takes two packets of salt flavoured chips?
Easy
- A9/38
- B10/20 × 9/19
- C10/20 × 10/19
- D1/4
8.Sonia has a bag containing 20 sweets. In the bag, there are 5 red, 6 green and 9 yellow sweets. Sonia chooses two sweets at random, without replacement, from the bag. What is the probability that she chooses two green sweets?
Easy
- A3/38
- B6/20 × 5/19
- C6/20 × 6/19
- D1/19

더 많은 문제

9.The heights of 120 boys are recorded. The table shows frequencies: | Height, h (m) | Frequency | |---|---| | 1.3 < h ≤ 1.4 | 7 | | 1.4 < h ≤ 1.5 | 18 | | 1.5 < h ≤ 1.6 | 30 | | 1.6 < h ≤ 1.7 | 24 | | 1.7 < h ≤ 1.8 | 27 | | 1.8 < h ≤ 1.9 | 14 | Three boys are chosen at random. What is the probability that one of the boys has a height greater than 1.8 m and the other two boys each have a height of 1.4 m or less?
Hard
- A(14/120)×(7/119)×(6/118)×3
- B(14/120)×(7/119)×(6/118)
- C(14/120)×(7/119)×(6/118)×6
10.200 students estimated the area of windows. The table shows frequencies: | Area, A (m²) | Frequency | |---|---| | 20 < A ≤ 60 | 32 | | 60 < A ≤ 100 | 64 | | 100 < A ≤ 150 | 80 | | 150 < A ≤ 250 | 24 | Two students are chosen at random from those that estimated the area to be more than 100 m². What is the probability that one of the two students estimates the area to be greater than 150 m² and the other estimates the area to be 150 m² or less?
Hard
- A(24/104)×(80/103)×2
- B(24/104)×(80/103)
- C(24/200)×(80/199)×2
- D(24/104)×(80/104)
11.A Venn diagram shows set A = {2,6,12,14,28}, set B = {7,14,21,28,35}, and universal set ξ = {1,2,5,6,7,12,14,21,28,35}. A number is picked at random from ξ. What is the probability that the number is in set A given that it is in set B?
Medium
- A2/5
- B2/10
- C1/5
- D3/5
12.A bag contains only red and blue marbles. There are 3 more red marbles than blue marbles. Kai takes 2 marbles from the bag at random. The probability that Kai takes 2 blue marbles is 1/7. How many blue marbles were in the bag initially?
Hard
- A6
- B3
- C9
- D12
13.Suleika has six cards numbered 1 to 6. She takes two cards at random without replacement. What is the probability that the sum of the numbers on the two cards is 5?
Medium
- A4/15
- B1/15
- C2/15
- D1/5
14.A box contains 20 packets of potato chips. 6 packets contain barbecue flavour, 10 packets contain salt flavour, 4 packets contain chicken flavour. Maria takes two packets at random without replacement. Show that the probability that she takes two packets of salt flavoured chips is 9/38.
Easy
- ATrue
- BFalse
- CCannot be determined
- D9/38 is correct
15.A bag contains 10 yellow beads, 6 blue beads and 4 green beads. A bead is taken at random and not replaced. A second bead is then taken. Find the probability that both beads are different colours.
Medium
- A62/95
- B33/95
- C248/380
- D1/3
16.Five cards are numbered 1, 2, 3, 4 and 5. Two of the cards are taken at random, without replacement. What is the probability that both cards show an even number?
Easy
- A1/10
- B2/5
- C1/5
- D3/10
17.Morgan picks two letters at random without replacement from the set {B, Y, M, A, T, H}. What is the probability that he picks the letter Y first?
Easy
- A1/6
- B1/5
- C1/36
- D1/30
18.Morgan picks two letters at random without replacement from the set {B, Y, M, A, T, H}. What is the probability that he picks the letter B then the letter Y?
Easy
- A1/30
- B1/36
- C1/15
- D1/6
19.A bag contains 4 red marbles and 2 yellow marbles. Behnaz picks two marbles at random without replacement. What is the probability that the marbles are not both red?
Medium
- A3/5
- B2/5
- C1/5
- D4/5
20.Six discs are numbered so that exactly two of them show the number 3. Two of the six discs are picked at random without replacement. Find the probability that both discs show the number 3.
Medium
- A1/15
- B2/15
- C1/6
- D1/3
21.Six discs are shared equally between three colours: 2 red, 2 blue and 2 green. Two of the six discs are picked at random without replacement. Find the probability that both discs are the same colour.
Medium
- A4/15
- B2/15
- C1/5
- D1/3
22.Morgan picks two letters at random without replacement from the set {B, Y, M, A, T, H}. What is the probability that he picks two letters that are the same?
Easy
- A0
- B1/30
- C1/6
- D1/15
23.Bag A contains 3 black balls and 2 white balls. Three balls are taken at random from bag A, without replacement. Find the probability that they are all black.
Easy
- A1/10
- B3/10
- C1/5
- D1/2
24.Bag A contains 3 black balls and 2 white balls. Three balls are taken at random from bag A, without replacement. Find the probability that they are all white.
Easy
- A0
- B1/10
- C1/5
- D1/2
25.Conditional probability is the probability of an event occurring given that another event has already occurred.
Easy
True or false?
26.A fair six-sided dice is rolled. Given that the number rolled is even, what is the probability that it is a multiple of 3?
Easy
- A1/6
- B1/3
- C1/2
- D2/3
27.A bag contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?
Medium
- A5/14
- B5/16
- C25/64
- D5/8
28.A bag contains 4 green and 6 yellow counters. Two counters are drawn without replacement. Which of the following probabilities are correct? (Select all that apply)
Medium
- AP(both green) = 2/15
- BP(both yellow) = 1/3
- CP(one of each colour) = 8/15
- DP(at least one green) = 2/3
29.Arrange the steps to find the probability that two cards drawn without replacement from a standard deck are both hearts.
Medium
- Multiply P(first heart) by P(second heart | first heart)
- Find P(second heart | first heart) = 12/51
- Find P(first heart) = 13/52 = 1/4
- Simplify product to get final probability
30.Match each description to the correct probability.
Medium
- P(A|B) when A and B are independent
- P(A|B) when A and B are mutually exclusive
- P(A|B) when A is a subset of B
- P(A∩B)/P(B)
- 0
- P(A)/P(B)
31.In a tree diagram for drawing without replacement, the probabilities on the second set of branches are the same regardless of the first outcome.
Easy
True or false?
32.A bag contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that the second marble is red given that the first marble is red?
Easy
- A4/7
- B5/8
- C4/8
- D5/7
33.A bag contains 10 yellow beads, 6 blue beads and 4 green beads. A bead is taken at random and not replaced. A second bead is then taken. What is the probability that both beads are the same colour?
Medium
- A\frac{232}{380}
- B\frac{248}{380}
- C\frac{132}{380}
- D\frac{62}{95}
34.In a class of 30 students, 18 study French, 12 study Spanish, and 6 study both. A student is chosen at random. Given that the student studies French, what is the probability they also study Spanish? Give your answer as a fraction in simplest form.
Easy
35.Complete the sentence: In a tree diagram for conditional probability, the probabilities on the branches from the same point must add up to ___.
Easy
In a tree diagram for conditional probability, the probabilities on the branches from the same point must add up to ____.
36.What is the notation for the probability of event A occurring given that event B has occurred? (Write using P, |, and brackets)
Easy
37.A bag contains only red and blue marbles. There are 2 more red marbles than blue marbles. Two marbles are drawn without replacement. The probability that both are blue is 2/15. Find the number of blue marbles.
Hard
38.A Venn diagram shows set A = {2,6,12,14,28}, set B = {7,14,21,28,35}, and universal set ξ = {1,2,5,6,7,12,14,21,28,35}. A number is picked at random from ξ. Given that the number is in set B, what is the probability that it is also in set A? Write your answer as a fraction in simplest form.
Medium
39.Complete the sentence about tree diagrams for conditional probability.
Easy
In a tree diagram for drawing without replacement, the probabilities on the second set of branches are ____ as the probabilities on the first set of branches.
40.Explain how to calculate P(A|B) using a two-way table.
Medium
41.Explain how to calculate the probability that two cards drawn without replacement from a standard deck of 52 cards are both hearts.
Medium
42.In a class of 30 students, 18 study French, 12 study Spanish, and 6 study both. A student is chosen at random. Given that the student studies French, what is the probability they also study Spanish? Give your answer as a fraction in simplest form.
Easy
- A1/3
- B1/2
- C3/5
- D2/3
43.A bag contains only red and blue marbles. There are 2 more red marbles than blue marbles. Two marbles are drawn without replacement. The probability that both are blue is 2/15. Find the number of blue marbles.
Hard
- A2
- B5
- C3
- D4

복습 노트

## Conditional Probability (IGCSE Extended) ### Key Idea A **conditional probability** is the probability of an event occurring *given* that another event has already happened. The sample space is reduced to only the outcomes of the given event. ### Calculating Conditional Probabilities - **Without replacement**: The number of outcomes decreases after each draw. - **Tree diagrams** help visualise successive events. - **Formula**: P(A given B) = number of outcomes in A that are also in B divided by total outcomes in B. ### Combined Conditional Probabilities When events happen one after another without replacement, multiply probabilities along branches of a tree diagram, adjusting the counts. #### Example: Drawing two marbles without replacement - P(blue first) = (number of blue) / (total) - P(blue second | blue first) = (blue left) / (total left) - P(both blue) = P(blue first) × P(blue second | blue first) ### Multiple Possibilities Use the **OR rule**: add probabilities of mutually exclusive outcomes. - For two events: P(different colours) = P(YB) + P(BY) + P(YG) + P(GY) + P(BG) + P(GB) - For three events: P(all same) = P(YYY) + P(BBB) + P(GGG) ### Algebraic Problems Set up an equation using the given probability, solve for the unknown quantity (e.g., number of marbles). ### Common Question Types - Drawing cards, marbles, or counters without replacement. - Selecting people from a group (e.g., from a table). - Using Venn diagrams or two-way tables with "given that" phrasing. ### Tips - Do **not** simplify fractions too early; keep common denominators for addition. - Remember that order matters: AB is different from BA. - "At least one" often means 1 − P(none).

Conditional Probability: The probability of an event occurring given that another event has already occurred. The sample space is restricted to the outcomes of the given event.
Without Replacement: When an item is selected and not returned to the set, so the total number of items and the number of items of each type decrease for subsequent selections.
Tree Diagram: A diagram used to show the probabilities of successive events, with branches representing possible outcomes and their probabilities.
OR Rule: The probability of either of two mutually exclusive events occurring is the sum of their individual probabilities.

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