Question

type the correct answer in each box.
in a game at an amusement park, a player can win 20 points, 25 points, or 30 points. the probability of winning these points is given in the table.

points	20	25	30

if jenny expects to win 22.5 points by playing the game once, the value of x is
and the value of y is

Explanation:

Step1: Recall Expected Value Formula

The expected value \( E(X) \) of a discrete random variable is given by \( E(X)=\sum x_iP(x_i) \), where \( x_i \) are the possible values and \( P(x_i) \) are their corresponding probabilities. Also, the sum of all probabilities must be 1, so \( 0.6 + x + y = 1 \), which implies \( x + y = 0.4 \).

Step2: Set Up Expected Value Equation

We know \( E(X) = 22.5 \), \( x_1 = 20, P(x_1)=0.6 \); \( x_2 = 25, P(x_2)=x \); \( x_3 = 30, P(x_3)=y \). So:

$$ 20\times0.6 + 25x + 30y = 22.5 $$

Simplify the left - hand side: \( 12+25x + 30y = 22.5 \), then \( 25x+30y=22.5 - 12=10.5 \).

Step3: Solve the System of Equations

We have the system:

1. \( x + y=0.4\Rightarrow y = 0.4 - x \)
2. \( 25x + 30y=10.5 \)

Substitute \( y = 0.4 - x \) into the second equation:

$$ 25x+30(0.4 - x)=10.5 $$

Expand: \( 25x + 12-30x = 10.5 \)
Combine like terms: \( - 5x=10.5 - 12=-1.5 \)
Solve for \( x \): \( x=\frac{- 1.5}{-5}=0.3 \)

Step4: Find the Value of y

Since \( y = 0.4 - x \) and \( x = 0.3 \), then \( y=0.4 - 0.3 = 0.1 \)

Answer:

The value of \( x \) is \( 0.3 \) and the value of \( y \) is \( 0.1 \)