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 as:

$E(S) = ext{sum of } (s imes P(S = s)) = 0 imes p + 1 imes 0.25 + 2 imes 0.20 + 4 imes 0.20 + 5 imes 0.20$ Substituting p:

$E(S) = 0 imes 0.05 + 1 imes 0.25 + 2 imes 0.20 + 4 imes 0.20 + 5 imes 0.20$

Calculating this results in:

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

To compute E(S²), we find:

$E(S^2) = ext{sum of } (s^2 imes P(S = s)) = 0^2 imes p + 1^2 imes 0.25 + 2^2 imes 0.20 + 4^2 imes 0.20 + 5^2 imes 0.20$ Substituting p:

$E(S^2) = 0^2 imes 0.05 + 1^2 imes 0.25 + 2^2 imes 0.20 + 4^2 imes 0.20 + 5^2 imes 0.20$

Calculating individually gives:

$= 0 + 0.25 + 0.80 + 3.20 + 5.00 = 9.45$

Using the formula for variance:

$Var(S) = E(S^2) - (E(S))^2$

We have already calculated:

$E(S^2) = 9.45$

And:

$E(S) = 2.45$

Thus, we compute:

$Var(S) = 9.45 - (2.45)^2 = 9.45 - 6.0025 = 3.4475$

Let's denote the probability of Jess winning as P(J). After two spins, Jess wins if she scores an odd number of times. The possible outcomes for her winning are:

1. Jess gets odd scores in both spins.
2. Jess gets an odd score in one spin and even in another.

Calculate P(J) based on the probabilities obtained previously (with p = 0.05) and total outcomes leading to Jess's win.

To find this probability, we calculate the scenarios where Tom wins exactly after three spins, ensuring that the even outcomes from spins are counted and odd outcomes do not lead to his win on the third spin. Using the appropriate probabilities, it can be calculated accordingly.

To find the probability of Jess winning after exactly 3 spins, we assess cases where Jess scores odd on the last spin and at least one odd on the first two spins. Probability calculations must consider individual outcomes against the winning criteria set out in the game rules.