The expected value is the sum of the probability of each result multiplied by the amount gained or lost.
In this case, is the probability of you getting two clubs is "p", the expected value is:
[tex]E=765\cdot p-46\cdot(1-p)[/tex]So, we just need to calculate "p".
The probability of getting the first club is 13 out of 52:
[tex]p_1=\frac{13}{52}=\frac{1}{4}=0.25[/tex]The probability of getting the second one is 12 out of 51, since there is one less club and on less card in the total.
[tex]p_2=\frac{12}{51}=0.23524\ldots[/tex]So, the probability of getting both is:
[tex]p=p_{1\cdot}p_2=0.25\cdot0.23524\ldots=0.0588235\ldots[/tex]Then, the expected value is:
[tex]\begin{gathered} E=765\cdot0.0588235\ldots-46(1-0.0588235) \\ E=45-43.2941\ldots=1.70588\ldots\approx1.71 \end{gathered}[/tex]So, the expected value is approximately $1.71.
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