The score when a spinner is spun is given by the discrete random variable X with the following probability distribution, where a and b are probabilities - Edexcel - A-Level Maths Statistics - Question 5 - 2018 - Paper 1

From our previous calculation, we have one linear equation:

$2a + 4b = 2$

Using the variance formula:

$Var(X) = E(X^2) - [E(X)]^2$

We compute:

$E(X^2) = (-1)^2 imes b + 0^2 imes a + 2^2 imes a + 4^2 imes a + 5^2 imes b$

This gives:

$E(X^2) = b + 0 + 4a + 16a + 25b = 26b + 20a$

Therefore, we have:

$Var(X) = (26b + 20a) - 2^2 = 26b + 20a - 4$

Given that Var(X) = 7.1, we set:

$26b + 20a - 4 = 7.1$

This simplifies to:

$26b + 20a = 11.1$

We have the system of equations:

1. $2a + 4b = 2$
2. $20a + 26b = 11.1$

From the first equation, solve for b:

$4b = 2 - 2a$ $b = 0.5 - 0.5a$

Substituting into the second equation:

$20a + 26(0.5 - 0.5a) = 11.1$

This simplifies to:

$20a + 13 - 13a = 11.1$ $7a = -1.9$ $a = -0.271$

Plugging a back into the expression for b:

$b = 0.5 - 0.5(-0.271) = 0.5 + 0.1355 = 0.6355$

The transformation of the random variable is given by:

$Y = 10 - 3X$

The expectation can be found by:

$E(Y) = E(10 - 3X) = 10 - 3E(X)$

Substituting E(X) = 2 gives:

$E(Y) = 10 - 3(2) = 10 - 6 = 4$

To calculate the variance of Y:

$Var(Y) = Var(10 - 3X) = (-3)^2 Var(X) = 9 Var(X)$

With Var(X) = 7.1,

$Var(Y) = 9 imes 7.1 = 63.9$

Using the transformation, we can set up the inequality:

$10 - 3X > X$

This simplifies to:

$10 > 4X$

Thus,

$X < 2.5$

From our probability distribution, we can find:

$P(Y > X) = P(X < 2.5) = P(X = -1) + P(X = 0) + P(X = 2) = b + a + a = b + 2a$

Substituting our earlier values of a and b gives us:

$P(Y > X) = 0.6355 + 2(-0.271) = 0.6355 - 0.542 = 0.0935$