You want to set up a games stall at the Winter Wonderland. You have a single pack of 52
cards and decide to play card games with your customers. The game is designed as follows. A
customer draws 4 cards at random. For every “Ace” that the customer draws, he/she wins £3
and for every face card (“Jack”, “Queen” or “King”), he/she wins £1.
Let the two discrete random variables U and V, be the number of aces and face cards
obtained, respectively.
(i) Derive the joint probability mass function p(U, V).
(ii) Find the marginal probability mass functions of U and V.
(iii) Are U and V independent? Specify the reason.
(iv) Find E(U), E(V), Var(U), Var(V) and Cov(U, V) and interpret the obtained values.
(v) Find the minimum price for the game, so that you don’t bear a loss if a very large
number of games are played.
(vi) If you set the price for each game by rounding up the value found in (v), find the
percentage of people who will go back home with positive winnings.
omg kabir is so gonna find this
To solve this problem, we will use basic concepts from probability and statistics.
(i) Deriving the joint probability mass function p(U, V):
Since the customer draws 4 cards at random from a pack of 52 cards, we can use the concept of combinations to determine the number of possible outcomes. There are a total of C(52, 4) = 270,725 possible ways to choose 4 cards from a deck of 52.
To find the joint probability mass function p(U, V), we need to calculate the probability of each possible pair (U, V) occurring.
Let's consider a particular pair (U, V) where U represents the number of aces and V represents the number of face cards obtained.
The number of ways to select U aces from 4 aces is C(4, U). Similarly, the number of ways to select V face cards from 12 face cards (4 jacks, 4 queens, and 4 kings) is C(12, V).
The number of ways to select the remaining 4 - U - V cards from the remaining 36 non-ace, non-face cards is C(36, 4 - U - V).
Therefore, the probability of getting U aces and V face cards is given by:
P(U, V) = (C(4, U) * C(12, V) * C(36, 4 - U - V)) / C(52, 4)
Repeat this computation for each possible pair (U, V) to obtain the complete joint probability mass function p(U, V).
(ii) Finding the marginal probability mass functions of U and V:
The marginal probability mass function of U, denoted as p(U), represents the probability distribution of the number of aces U, regardless of the value of V. Similarly, the marginal probability mass function of V, denoted as p(V), represents the probability distribution of the number of face cards V, regardless of the value of U.
To calculate p(U), you sum up the joint probabilities p(U, V) for all possible V values:
p(U) = ∑ p(U, V) for each V
Similarly, to calculate p(V), you sum up the joint probabilities p(U, V) for all possible U values:
p(V) = ∑ p(U, V) for each U
(iii) Determining whether U and V are independent:
Two random variables U and V are independent if and only if their joint probability mass function p(U, V) is equal to the product of their marginal probability mass functions p(U) and p(V).
Therefore, if p(U, V) = p(U) * p(V) for all possible pairs (U, V), then U and V are independent. Otherwise, they are dependent.
(iv) Calculating E(U), E(V), Var(U), Var(V), and Cov(U, V):
To calculate the expected value (mean) E(U) and E(V), you multiply each possible value of U and V by their respective probabilities p(U) and p(V), and then sum them up.
The variance Var(U) can be calculated as E((U - E(U))^2) using the expected value E(U). Similarly, the variance Var(V) can be calculated as E((V - E(V))^2) using the expected value E(V).
The covariance Cov(U, V) measures the linear relationship between U and V. It can be calculated as E((U - E(U)) * (V - E(V))) using the expected values E(U) and E(V).
(v) Finding the minimum price for the game:
To set the minimum price for the game, we need to ensure that on average, the earnings from the game cover the cost. In this case, the cost is 0 because the game is played with a single pack of 52 cards.
The expected earnings per game can be calculated as Earnings = 3 * E(U) + 1 * E(V). Set this equal to 0 and solve for the minimum price.
(vi) Finding the percentage of people with positive winnings:
To find the percentage of people who go back home with positive winnings, calculate the probability of winning something (either £3 or £1) in a single game. This probability can be calculated as the sum of probabilities p(U, V) for all pairs (U, V) where U > 0 or V > 0.
Divide this probability by the total probability of playing the game (sum of probabilities p(U, V) for all pairs (U, V)) and multiply by 100 to get the percentage.