Answer:
Expect to lose about $0.73 or 73 cents for each game played
Step-by-step explanation:
Let's define the four events:
F = event of drawing a face card
N = event of drawing a non-face card
H = event of the coin landing on heads
T = event of the coin landing on tails
The events F and N are complementary and this means that one event or the other, but not both, must happen. We either draw a face card (F) or we don't (N). This is why the probabilities add to 1
P(F) + P(N) = 1
Thus;
P(N) = 1 - P(F)
There are 4 suits with 3 face cards per suit (King, Queen, Jack).
So 4 x 3 = 12 face cards out of 52 cards total.
Therefore,
P(F) = probability of drawing a face card = (number of face cards)/(number of cards total)
P(F) = 12/52 = 3/13
And,
P(N) = probability of drawing a non-face card
P(N) = 1 - P(F)
P(N) = 1 - (3/13)
P(N) = 10/13
Now, assuming we have a fair coin with either side is likely to be landed on, it means that;
P(H) = 1/2
P(T) = 1/2
So, P(H) + P(T) = 1
Assuming the events of drawing a card and flipping a coin are independent, then we can form the compound probabilities
P(F & H) = P(F) x P(H)
P(F & H) = (3/13) x (1/2) = 3/26
P(F & T) = P(F) x P(T)
P(F & T) = (3/13)*(1/2) = 3/26
Now, Similar to the probability P(X) notation, let's introduce the function V(X) where V is the net value and X is the general event. To be more specific, writing V(F) represents the net value of drawing a face card.
The three cases we're concerned with are:
V(F & H) = net value for getting face card and heads = 5
V(F & T) = net value for getting face card and tails = 2
V(N) = net value for getting non face card = -2
The negative value (-2) indicates a loss of 2 dollars.
When we play the game out, there are three cases:
Case A = drawing a face card and the coin landing on heads
Case B = drawing a face card and the coin landing on tails
Case C = drawing a non-face card
What we do is multiply the probabilities for each case happening with the net values for each case.
Thus;
For case A, we have the probability P(F & H) = 3/26 and the net value V(F & H) = 5
Hence;
P(F & H) x V(F & H) = (3/26) x 5 = 15/26 = 15/26
Similarly for case B
P(F & T) x V(F & T) = (3/26) x 2 = 6/26 = 3/13
and finally case C
P(N) x V(N) = (10/13) x (-2) = -20/13
Let's now add them up to get;
(15/26) + (3/13) + (-20/13)
This gives; (15 + 6 - 40)/26 = -19/26 = $-0.73
At this expected value, it means that we expect to lose about $0.73 or 73 cents for each game played. This is not a fair game (because expected value isn't 0). Thus, the game clearly favors the house instead of the player.
