A spinner is designed so that the score S is given by the following probability distribution - Edexcel - A-Level Maths Statistics - Question 8 - 2011 - Paper 2

The expected value E(S) is calculated using:

$E(S) = 0 imes p + 1 imes 0.25 + 2 imes 0.20 + 4 imes 0.20 + 5 imes 0.20$

Substituting the value of p:

$E(S) = 0 imes 0.15 + 1 imes 0.25 + 2 imes 0.20 + 4 imes 0.20 + 5 imes 0.20$

Calculating gives:

$E(S) = 0 + 0.25 + 0.40 + 0.80 + 1.00 = 2.45$.

To find E(S²), we use the formula:

$E(S²) = 0² imes p + 1² imes 0.25 + 2² imes 0.20 + 4² imes 0.20 + 5² imes 0.20$

Substituting the value of p:

$E(S²) = 0 imes 0.15 + 1 imes 0.25 + 4 imes 0.20 + 16 imes 0.20 + 25 imes 0.20$

Calculating gives:

$E(S²) = 0 + 0.25 + 0.80 + 3.20 + 5.00 = 9.45$.

The variance Var(S) can be calculated using:

$Var(S) = E(S²) - (E(S))²$

Since we have already found that E(S) = 2.45 and E(S²) = 9.45, we substitute these values:

$Var(S) = 9.45 - (2.45)²$

Calculating yields:

$Var(S) = 9.45 - 6.0025 = 3.4475$.

To find the probability that Jess wins after 2 spins, we consider the possible outcomes:

• On both spins, if at least one is odd, Jess wins. The calculations for this would involve finding the total probability of odd scores across the two spins. Let P(odd) be the sum of probabilities of getting an odd score: $P(odd) = p + 0.25 = 0.15 + 0.25 = 0.40$. The probability that Jess wins after 2 spins is: $P(Jess ext{ wins}) = 1 - P(Tom ext{ wins}) = 1 - P(odd ext{ on both}) = 1 - (0.40)^2 = 1 - 0.16 = 0.84$.

To find the probability that Tom wins after exactly 3 spins, we first calculate the total probability that Tom wins on the third spin. This involves considering that Tom must not win on the first two spins but must win on the third. This requires detailed calculations for the probabilities involved: Count cases of odd/even scores across the spins.

Assuming P(Tom wins) = 0.60: $P(Tom ext{ wins after 3 spins}) = P( ext{Tom loses first two}) imes P( ext{Tom wins third}) = 0.40^2 imes 0.60 = 0.096$.

For Jess to win after exactly 3 spins, she must be odd for the winning side on the third spin specifically while not winning in the first two spins. This gives: $P(Jess ext{ wins after 3}) = (P(odd)^2)(P(Jess ext{ win on third})) = (0.40^2) imes (0.40) = 0.064$.