Conditional Probability

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?

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?

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?

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?

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?

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.

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.

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?

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?

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?

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?

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.

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.

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?

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.

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.

25.Conditional probability is the probability of an event occurring given that another event has already occurred.

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?

27.A bag contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?

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)

29.Arrange the steps to find the probability that two cards drawn without replacement from a standard deck are both hearts.

30.Match each description to the correct probability.

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.

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?

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?

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.

35.Complete the sentence: 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)

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.

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.

39.Complete the sentence about tree diagrams for conditional probability.

40.Explain how to calculate P(A|B) using a two-way table.

41.Explain how to calculate the probability that two cards drawn without replacement from a standard deck of 52 cards are both hearts.

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.

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.