1. A jar contains five blue balls and five red balls. You roll a fair die once. Next you randomly draw (without replacement) as many balls from the jar as the number of points you have rolled with the die.
(a) What is the probability that all of the balls drawn are blue?
(b) What is the probability that the number of points shown by the die is r given that all of the balls drawn are blue?
P(n) = 1/6
Clearly, P(6 blues) = 0
P(n blues) = 5*4*3*.../(1*2*...n) = 5Cn
so P(n & n blues) = 5Cn/6 for 0<n<6
oops. I got the denominator in the wrong order.
P(n) = 5/6 * 4/5 * ... n factors = 5*4*3.../(6*5*4...)
= 5Pn*(6-n)!/6!
To determine the probability in this situation, we need to understand the total number of possible outcomes and the number of favorable outcomes.
(a) Let's calculate the probability that all of the balls drawn are blue.
Total number of ways to roll a fair die: 6 (since it has 6 faces numbered 1 to 6)
Total number of ways to draw balls from the jar: 10 (since there are 10 balls in total)
In order to draw all blue balls, we need to roll a die and draw the same number of balls as the number on the die. So, we need to consider each possible outcome of the die roll.
Number of outcomes where all balls drawn are blue:
- If we roll a 1, we can choose any 1 of the 5 blue balls.
- If we roll a 2, we can choose any 2 of the 5 blue balls.
- If we roll a 3, we can choose any 3 of the 5 blue balls.
- If we roll a 4, we can choose any 4 of the 5 blue balls.
- If we roll a 5, we can choose any 5 of the 5 blue balls.
- If we roll a 6, we can choose any 5 of the 5 blue balls.
So, in total, there are 1 + 5 + 10 + 10 + 5 + 1 = 32 possible ways to draw all blue balls.
Therefore, the probability that all of the balls drawn are blue is the number of favorable outcomes (32) divided by the total number of possible outcomes (6 x 10 = 60):
P(all blue) = 32/60 = 8/15 ≈ 0.5333
(b) Now let's calculate the probability that the number of points shown by the die is r given that all the balls drawn are blue.
Since we already know that all of the balls drawn are blue, we only need to consider the number of points shown by the die. In this case, the number of points shown by the die can be any integer from 1 to 5, because rolling a 6 wouldn't allow us to draw all blue balls.
Number of outcomes where the number of points shown by the die is r: 1 (since there is only one way to draw r balls, where r ranges from 1 to 5)
Therefore, the probability that the number of points shown by the die is r given that all the balls drawn are blue is:
P(number of points is r | all blue) = 1/32 (since there are 32 possible ways to draw all blue balls)
Please note that in both scenarios, we assumed that the balls were drawn without replacement, meaning that once a ball is drawn, it is not put back into the jar before the next draw.