Coin Flipping - Probability and Value Proposition

Question

Rusty with probability here...

The question is: Flip a coin 11 times. If you get 8 tails or less, I will pay you \$1.
Otherwise, you pay me \$7.

Step 1. Find the expected value of the proposition. Round your answer to two decimal places.

Step 2. If you played this game 615 times how much would you expect to win or lose? (Losses must be entered as negative.)

I think that the expected value of the proposition is the sum of the probability of getting 8 tails or less times times the 1 dollar value, plus probability of getting more thand 8 tails times the $7 value.

I would like a walkthrough of the mindset and reasoning, not just the answer so I can learn how to approach the problem. I am also learning how to solve with R.

Answer 1

There are $2^{11}$ possible outcomes, of these there are $\binom{11}{9}+\binom{11}{10}+\binom{11}{11}=67$ outcomes in which I pay you $7$ dollars.

Therefore the expected cash the other player gets is:

$\frac{2^{11}-67}{2^{11}}-7\frac{67}{2^{11}}=\frac{1512}{2^{11}}\approx0.73$.

So I don't recommend you play the game.

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The expected cash after playing the game $n$ times is $0.73\times n$

Answer 2

Let $X$ be the random variable representing the number of tails in $11$ tosses of a fair coin. We have

$$P(X > 8) = \sum_{k=9}^{11} \binom{11}{k}\frac{1}{2^k}\frac{1}{2^{11-k}}$$

This can be derived by noting that for this distribution, we have 11 spots to place a metaphorical heads or a tails. Each spot has a $\frac{1}{2}$ chance of being a heads or a tails, and we have $11$ choose $k$ ways of rearranging these spots around.

Let $Y$ be the random variable representing the amount of money we make from playing the game once. We have

$$E(Y) = (1)(1 - P(X>8)) - 7P(X>8)$$ $$E(Y) = 1 - 8\sum_{k=9}^{11} \binom{11}{k}\frac{1}{2^k}\frac{1}{2^{11-k}}$$

$$E(Y) = \frac{189}{256}$$

Playing this game 615 times we expect to have made

$$615 * \frac{189}{256} = \frac{116235}{256}$$

To reiterate in decimals, we expect to make roughly $74$ cents each time we play the game and if we play it 615 times, we expect to make roughly $ \$ 454.04$ .

Note: I have answered this taking the perspective that the "you" in the question is the "we" in my answer