Laura's school is holding an outdoor activities celebration event for its 20th anniversary. Laura decides to participate in the Best Striker football game. In this game, the player shoots at goal which is split into 6 regions assigned with the scores, X, as shown in the diagram below. 2 1 4 -3 0 If the shot goes outside of the mini goal player receives a score of O. The following table shows the probability distribution for X 2 4 -1 Score (X) -3 3 12 0 4 12 1 2 12 1 12 P 1 12 Pr(X =) [1] 121 (a) Find the exact value of p. Laura plays the game once. (b) Calculate the expected score. Laura plays the game twice. (e) Find the probability that she has a total score of 5 after two rounds.

The value of p is 1/12
The expected score of the game is -1/6
The probability that she has a total score of 5 after two rounds is 0

To calculate the value of p, we make use of the following probability formula

[tex]\sum P(x) = 1[/tex]

So, we have:

[tex]\frac 3{12} + \frac 1{12} + \frac 4{12} + \frac 2{12} + p + \frac 1{12} = 1[/tex]

Collect like terms

[tex]\frac 3{12} + \frac 1{12} + \frac 4{12} + \frac 2{12} + \frac 1{12} + p= 1[/tex]

Evaluate the sums, on the left-hand side

[tex]\frac{11}{12} + p = 1[/tex]

Subtract 11/12 from both sides

[tex]p = \frac 1{12}[/tex]

Hence, the value of p is 1/12

This is calculated using the following expected value formula

[tex]E(x) = \sum x * P(x)[/tex]

So, we have:

[tex]E(x) = -3 * \frac{3}{12} -1 * \frac 1{12} + 0 * \frac 4{12} + 1 * \frac 2{12} + 2 * \frac 1{12} + 4 * \frac 1{12}[/tex]

Evaluate the products

[tex]E(x) = -\frac{9}{12} - \frac 1{12} + \frac 2{12} + \frac 2{12} + \frac 4{12}[/tex]

[tex]E(x) = -\frac 2{12}[/tex]

Simplify

[tex]E(x) = -\frac 1{6}[/tex]

Hence, the expected score of the game is -1/6

From the table, we have:

[tex]P(5) = 0[/tex]

For after two rounds, we make use of the following equation

[tex]P(5) * P(5) = 0 * 0[/tex]

[tex]P(5) * P(5) = 0[/tex]

Hence, the probability that she has a total score of 5 after two rounds is 0