In a gambling game a man is paid $5 if he gets all heads or all tails when 3 coins are tossed, and he pays out of $3 if either 1 or 2 heads show. what is his expected gain or loss of the 3 coins?

There are 8 possible outcomes 2^3

Since you either have a head or a tail and 3 coins.

2/8 or 1/4 HHH or TTT

6/8 or 3/4 for the other choices.

1/4 times 5 - 3/4 times 3

-$1

Well, if we think about it, this man is basically paying to play a game called "Heads or Tails or Kinda Both."

Let's calculate his expected gain or loss. First, we need to know the probability of each outcome.

There are 8 possible outcomes when tossing 3 coins:

1. HHH (all heads)
2. HHT (2 heads, 1 tail)
3. HTH (2 heads, 1 tail)
4. HTT (1 head, 2 tails)
5. THH (2 heads, 1 tail)
6. TTH (1 head, 2 tails)
7. THT (1 head, 2 tails)
8. TTT (all tails)

Outcomes 1 and 8 result in a gain of $5, while outcomes 2, 3, 4, 5, 6, and 7 result in a loss of $3.

The probability of getting all heads or all tails is 2/8 = 1/4 since there are two favorable outcomes out of eight.

The probability of getting 1 or 2 heads is 6/8 = 3/4 since there are six favorable outcomes out of eight.

Now, we calculate the expected gain or loss:

Expected gain/loss = (Probability of gain × Amount gained) + (Probability of loss × Amount lost)
= (1/4 × $5) + (3/4 × -$3)
= $1.25 - $2.25
= -$1

So, the man's expected gain or loss of the 3 coins is -$1. Therefore, he can expect to lose $1 on average every time he plays the game.

But hey, on the bright side, the man might stumble upon that elusive outcome where all the coins land on their edges! Then he could ask for a bonus... or start a circus act! 🤡

To calculate the man's expected gain or loss, we need to determine the probability of each outcome and multiply it by the associated gain or loss.

Let's consider the possible outcomes of tossing 3 coins:

1. All heads (HHH): The probability of getting all heads is 1/2 * 1/2 * 1/2 = 1/8. The man gets $5, so his gain is +$5.

2. All tails (TTT): The probability of getting all tails is also 1/8. The man gets $5, so his gain is also +$5.

3. 1 Head (HHT, HTH, or THH): The probability of getting exactly 1 head is 3/8. The man pays out $3, so his loss is -$3.

4. 2 Heads (HTH, HHT, or THH): The probability of getting exactly 2 heads is also 3/8. The man pays out $3, so his loss is again -$3.

To calculate the expected gain or loss, we multiply each outcome's probability by its associated gain or loss, and then sum the results:

Expected gain/loss = (P(all heads) * gain from all heads) + (P(all tails) * gain from all tails) + (P(1 head) * loss from 1 head) + (P(2 heads) * loss from 2 heads)
= (1/8 * $5) + (1/8 * $5) + (3/8 * -$3) + (3/8 * -$3)
= $5/8 - $3/8 - $9/8 - $9/8
= -$16/8
= -$2

Therefore, the man's expected gain or loss from the gambling game is -$2.

To determine the man's expected gain or loss, we need to calculate the probability of each possible outcome and then multiply it by the corresponding gain or loss for that outcome.

Let's calculate the probability for each outcome:

1. Probability of all heads: When three coins are tossed, the probability of getting all heads is (1/2)^3 = 1/8, since each coin has a 1/2 chance of coming up heads.
2. Probability of all tails: Similarly, the probability of getting all tails is also 1/8.
3. Probability of 1 or 2 heads: To calculate this probability, we need to subtract the probabilities of both all heads and all tails from 1. So, the probability of getting 1 or 2 heads is 1 - (1/8 + 1/8) = 6/8 = 3/4.

Now that we have the probabilities for each outcome, let's calculate the gain or loss for each:

1. All heads: The man gets $5.
2. All tails: Again, the man gets $5.
3. 1 or 2 heads: The man pays out $3.

Now we can calculate the man's expected gain or loss using the following formula:

Expected Gain or Loss = (Probability of Outcome 1 x Gain/Loss for Outcome 1) + (Probability of Outcome 2 x Gain/Loss for Outcome 2) + ...

Substituting the values we calculated:

Expected Gain or Loss = (1/8 x $5) + (1/8 x $5) + (3/4 x -$3)
Expected Gain or Loss = $5/8 + $5/8 - $9/4
Expected Gain or Loss = $10/8 + $10/8 - $36/8
Expected Gain or Loss = $20/8 - $36/8
Expected Gain or Loss = -$16/8
Expected Gain or Loss = -$2

Therefore, the man's expected gain or loss of the 3 coins is -$2, which means he is expected to lose $2 on average.