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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.
\begin{tabular}{|c|c|}
\hline Points & Probability \\
\hline 20 & 0.6 \\
\hline 25 & [tex]$x$[/tex] \\
\hline 30 & [tex]$y$[/tex] \\
\hline
\end{tabular}

If Jenny expects to win 22.5 points by playing the game once, the value of [tex]$x$[/tex] is [tex]$\square$[/tex] and the value of [tex]$y$[/tex] is [tex]$\square$[/tex].



Answer :

GinnyAnswer
Let's denote the following:

- Points won: 20 points, 25 points, 30 points
- Their probabilities: 0.6, [tex]\( x \)[/tex], [tex]\( y \)[/tex]

We are given that the expected number of points is 22.5. Therefore, we can set up the equation for the expected value as follows:

[tex]\[ \text{Expected value} = (20 \times 0.6) + (25 \times x) + (30 \times y) = 22.5 \][/tex]

We are also given that the sum of all probabilities must equal 1:

[tex]\[ 0.6 + x + y = 1 \][/tex]

First, let's solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:

[tex]\[ y = 1 - 0.6 - x \][/tex]
[tex]\[ y = 0.4 - x \][/tex]

Next, substitute [tex]\( y \)[/tex] into the expected value equation:

[tex]\[ 20 \times 0.6 + 25 \times x + 30 \times (0.4 - x) = 22.5 \][/tex]

Calculate [tex]\( 20 \times 0.6 \)[/tex]:

[tex]\[ 12 \][/tex]

Substitute the value into the equation:

[tex]\[ 12 + 25x + 30(0.4 - x) = 22.5 \][/tex]

Distribute the 30:

[tex]\[ 12 + 25x + 12 - 30x = 22.5 \][/tex]

Combine like terms:

[tex]\[ 24 - 5x = 22.5 \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ -5x = 22.5 - 24 \][/tex]
[tex]\[ -5x = -1.5 \][/tex]
[tex]\[ x = 0.3 \][/tex]

Now, substitute the value of [tex]\( x \)[/tex] back into the equation for [tex]\( y \)[/tex]:

[tex]\[ y = 0.4 - 0.3 \][/tex]
[tex]\[ y = 0.1 \][/tex]

The values are:

[tex]\[ x = 0.3 \][/tex]
[tex]\[ y = 0.1 \][/tex]

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