I need help finding the expected value for this game

Question

A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

1. If the card is a face card, and the coin lands on Heads, you win $4
2. If the card is a face card and the coin lans on Tails, you win $1
3. If the card isn't a face card, I lose $1, no matter what the coin shows

I need help finding the expected value for this game (expected net gain or loss).If you could explain it step by step it would help a lot. I also have to round the answer to two decimal places.

Answer 1

Hey Mikayla,

For some random variable X with possible values x₁, x₂, ... , x_n, the expected value, E[X], is the probability-weighted sum of possible values. In other words, the expected value is the result of adding up each possible outcome multiplied by the probability that particular outcome occurs.

In your problem, there are three random variables: the card drawn from the 52-card deck, the result of the coin flip, and the money we win. We know the card draw and coin flip are independent of each other while the money won is dependent on the combination the two.

Let the random variable X = {4, 1, -1} denote the money won. To find the amount of money we expect to win, plug this into our earlier definition:

E[X] = (p₁)(4) + (p₂)(1) +(p₃)(-1)

Now, all that's left to do is calculate the probabilities, plug em in, and add it all up!

p₁ = P(face card, heads) = P(face card)⋅P(heads)

p₂ = P(face card, tails) = P(face card)⋅P(tails)

p₃ = P(not a face card)

Hope this helps! Lemme know if you have any other questions.