There are 3 red balls and 7 blue balls in Bowl A, and 8 red balls and 2 blue balls in Bowl
B. Mandy rolls a die and if a prime number is up, Bowl A is selected; otherwise Bowl B is
selected. Then Katie draws a ball randomly from the selected bowl. Find the probability
of Bowl A being selected given that a red ball is drawn.
To find the probability of Bowl A being selected given that a red ball is drawn, we can use Bayes' Theorem.
Let's break down the problem step by step:
Step 1: Calculate the probability of selecting Bowl A and drawing a red ball.
The probability of selecting Bowl A is 3/6, which can be simplified to 1/2 since there are only two possible bowls (A and B) to choose from. The probability of drawing a red ball from Bowl A is 3/10, as there are 3 red balls and 10 balls in total in Bowl A.
So, the probability of selecting Bowl A and drawing a red ball is (1/2) * (3/10) = 3/20.
Step 2: Calculate the probability of selecting Bowl B and drawing a red ball.
The probability of selecting Bowl B is also 1/2 since there are only two possible bowls (A and B). The probability of drawing a red ball from Bowl B is 8/10, as there are 8 red balls and 10 balls in total in Bowl B.
So, the probability of selecting Bowl B and drawing a red ball is (1/2) * (8/10) = 8/20.
Step 3: Calculate the probability of drawing a red ball from any bowl.
The probability of drawing a red ball can be found by considering the total number of red balls and total number of balls overall from both bowls. In Bowl A, there are 3 red balls, and in Bowl B, there are 8 red balls. So, the total number of red balls is 3 + 8 = 11. The total number of balls overall is 10 + 10 = 20 (10 balls in each bowl).
Hence, the probability of drawing a red ball from any bowl is 11/20.
Step 4: Use Bayes' Theorem to calculate the probability of selecting Bowl A given that a red ball is drawn.
Bayes' Theorem states that P(A|B) = (P(B|A) * P(A)) / P(B), where P(A|B) represents the probability of event A occurring given that event B has occurred, P(B|A) represents the probability of event B occurring given that event A has occurred, P(A) represents the probability of event A occurring, and P(B) represents the probability of event B occurring.
In our case, A represents selecting Bowl A, and B represents drawing a red ball.
So, the probability of selecting Bowl A given that a red ball is drawn, P(A|B), can be calculated as follows:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (3/20) / (11/20)
= (3/20) * (20/11)
= 3/11
Therefore, the probability of Bowl A being selected given that a red ball is drawn is 3/11.