Question

Consider a game in which players draw playing cards one at a time from a standard 52-card deck. If a player draws a face card (a jack, a queen, or a king), the player is awarded 16 points. Any other card drawn earns the player 3 points. What is the expected value of drawing a card in this game on the first turn?
A. 3.7 points B. 6 points C. 6.3 points D. 9.5 points

Answer 1

Answer:

Expected value of drawing a card in this game on the first turn = 6

Step-by-step explanation:

Given - Consider a game in which players draw playing cards one at a time from a standard 52-card deck. If a player draws a face card (a jack, a queen, or a king), the player is awarded 16 points. Any other card drawn earns the player 3 points.

To find -  What is the expected value of drawing a card in this game on the first turn?

Formula used -

Expected value, E[x] = ∑ x p(x)

where p(x) is the probability

Proof -

Total cards in a standard deck = 52

Total face cards = 12 (a jack, a queen, or a king)

Other cards = 40

Now,

Probability of getting a face card = $\frac{12}{52}$

Probability of getting a other card = $\frac{40}{52}$

So,

Expected value, E[x] = ∑ x p(x)

                                  = (16)($\frac{12}{52}$) + (3)($\frac{40}{52}$)

                                  = $\frac{192}{52} + \frac{120}{52}$

                                  = $\frac{312}{52}$

                                  = 6

∴ we get

Expected value of drawing a card in this game on the first turn = 6

So,

The correct option is - B. 6 points