Math Problem

Question

Assume that the box contains 8 balls: 2 red, 2 blue, and 4 yellow. As in the text, you draw one ball, note its color, and if it is yellow replace it. If it is not yellow you do not replace it. You then draw a second ball and note its color.

 

(1) What is the probability that the second ball drawn is yellow?

 

(2) What is the probability that the second ball drawn is red?

Answer 1

Part (1) Probability that second ball drawn is yellow = 

 

Probability that first ball drawn is yellow * probability that second ball drawn is yellow + probability that first ball drawn is not yellow * probability that second ball drawn is yellow 

 

= (4/8) * (4/8) + (4/8) * (4/7)

 

= (1/4) + (2/7)

 

= 15/28

 

Explanation: There are 2 scenarios:

 

In scenario 1, you will get yellow ball in first draw and probability of getting a yellow is (4/8) in this instance, you will replace the ball and again pick a ball hence in second draw as well you will have equal probability of getting a yellow ball = 4/8 

hence probability of this scenario is (4/8) * (4/8)

 

Scenario 2, you will not get yellow ball in first draw, probability of this is (4/8), in this case you do not replace the ball hence you have one less yellow ball in the box and thus probability of getting second yellow ball is (4/7) since we are left with 4 yellow balls out of 7 total balls

 

(2) Probability of second ball drawn is red (with same logic as above)

 

Scenario 1: Probability of first ball drawn in yellow * probability of second ball drawn in red 

 

= (4/8) * (2/8)

 

Scenario 2: probability of first ball drawn is red * probability of second ball drawn is red

 

=  (2/8) * (1/7)

 

Scenario 3: Probability of first ball drawn is blue * probability of second ball drawn is red

 

= (2/8) * (2/7)

 

Probability of second ball drawn is red = 

 

(4/8) * (2/8) + (2/8) * (1/7) + (2/8) * (2/7)

 

= 13/56