Question

You are playing a game by drawing a card from a standard deck and replacing it.If the card is face card, you win $\backslash (30$.If it is not face card,you pay $\backslash )2$.There are $12$face card in a deck of $52$cards.What is the expected value of playing game?

Short answer

The expected value of playing game is$5.392$.

Step-by-step solution

Calculation of probabilities

Calculate probability:

There are $52$cards.

The no of facing card is$12$.

So,

If the card is face card,

$\mathrm{P}\left(\mathrm{x}\right)=\frac{\mathrm{total}\mathrm{no}\mathrm{of}\mathrm{facing}\mathrm{card}}{\mathrm{total}\mathrm{no}\mathrm{of}\mathrm{cards}}$

$\mathrm{P}\left(\mathrm{x}\right)=\frac{12}{52}$

$=0.231$

If the card is not facing card,

Number of not facing card is $40$.

So,

$\mathrm{P}\left(\mathrm{x}\right)=\frac{\mathrm{total}\mathrm{no}\mathrm{of}\mathrm{not}\mathrm{facing}\mathrm{card}}{\mathrm{total}\mathrm{no}\mathrm{of}\mathrm{cards}}$

$=\frac{40}{52}$

$=0.769$

The values are shown in following table.

Calculation for expected value

The value for expected is,

$\mathrm{\mu }=0.231\times 30-2\times 0.769$

$=6.93-1.538$

$=5.392$