I have to solve this problem:
Three urns each contain ten gems:
- Urn 1 contains 6 rubies and 4 emeralds.
- Urn 2r contains 8 rubies and 2 emeralds.
- Urn 2e contains 4 rubies and 6 emeralds.
The following procedure is used to select two gems. First, one gem is drawn at random from urn 1. If this first gem is a ruby, then a second gem is drawn at random from urn 2r; however, if the first gem is an emerald, then the second gem is drawn at random from urn 2e.
Suppose that this procedure is independently replicated three times. What is the probability that a ruby is obtained on the second draw exactly once?
My solution was to calculate the probability of obtaining a ruby on the second draw, where I got $0.64$ which is also correct according to the solutions.
Then to solve the actual problem I wanted to use R, writing
pbinom(1,3,0.64) - pbinom(0,3,0.64)
, getting $0.248832$ as an answer.
Yet the correct solution would be $0.3549$.