Probability work

Question

An experiment consists of selecting a ball from a bag and spinning a coin. The bag contains 5 red balls and 7 blue balls. A ball is selected at random from the bag, it's colour is noted and then the ball is returned to the bag. When a red ball is selected, a biased coin with probability ⅔ of landing heads is spun. When a blue ball is selected, a fair coin is spun.

Shivani selects a ball and spins the appropriate coin.

1. Find the probability that she obtains a head.
2. Given that Tom selected a ball at random and obtained a head when he spun the appropriate coin, find the probability that Tom selected a red ball.
3. Shivani and Tom each repeat this experiment, Find the probability that the color of the ball Shivani selects is the same as the color of the ball Tom selects

Answer 1

Denote the following letters and symbol:

H = coin lands on head

R = red ball is selected

B = blue ball is selected

∩ = intersection

1) We want to find P(H).

1. Rewrite P(H) = P(H ∩ R) + P(H ∩ B)
2. Then use Bayes Rule:
3. P(H ∩ R) = P(H | R) * P(R) and P(H ∩ R) = P(H | B) * P(B)
4. We know the following probabilities from the problem:
5. P(H | R) = 2/3, P(H | B) = 1/2, P(R) = 5/12, P(B) = 7/12

2) We want to find P(R | H)

1. Use Bayes Rule:
2. P(R | H) = P(H | R) * P(R) / P(H)

Use the probabilities from #1 to find this answer

3) We want to find P(B ∩ B) + P(R ∩ R)

1. Assuming independence, P(B ∩ B) = P(B)*P(B) and P(R ∩ R) = P(R)*P(R)

Use the probabilities from #1 to find this answer