A coin is tossed three times. If heads appears on all 3 tosses, Mary will win $16. If heads appears on 2 of the

tosses, she will win $2. The game costs $5 to play. What is her mean expectation?

To calculate Mary's mean expectation, we need to calculate the probability of each outcome and multiply it by the amount she wins or loses.

Let's calculate the probability for each outcome:

1. Probability of getting heads on all 3 tosses: When a fair coin is tossed, the probability of getting heads is 1/2. Since the coin is tossed 3 times independently, the probability of getting heads on all 3 tosses is (1/2) * (1/2) * (1/2) = 1/8.

2. Probability of getting heads on 2 of the tosses: There are 3 ways this can happen: HHT, HTH, or THH. Each arrangement has a probability of (1/2) * (1/2) * (1/2) = 1/8. So, the total probability of getting heads on 2 tosses is 3 * (1/8) = 3/8.

3. Probability of getting heads on fewer than 2 tosses: This is the complement of the previous two outcomes. Since there are no other possibilities, the probability of getting heads on fewer than 2 tosses is 1 - ((1/8) + (3/8)) = 4/8 = 1/2.

Now, let's calculate the total win or loss for each outcome:

1. If heads appears on all 3 tosses, Mary wins $16. So, her total win is $16.

2. If heads appears on 2 of the tosses, Mary wins $2. So, her total win is $2.

3. If heads appears on fewer than 2 tosses, Mary loses $5 (the cost of playing the game). So, her total loss is -$5.

Finally, let's calculate the mean expectation by multiplying each outcome by its probability and summing the results:

Mean expectation = (1/8) * $16 + (3/8) * $2 + (1/2) * (-$5)
= $2 + $0.75 - $2.5
= $0.25

Therefore, Mary's mean expectation is $0.25.

To calculate Mary's mean expectation, we need to determine the probability of each possible outcome and then multiply it by the corresponding amount she would win or lose.

Let's break down the possible outcomes:

1. If heads appears on all 3 tosses:
- The probability of getting heads on a single toss is 1/2 (assuming a fair coin).
- Since there are 3 independent tosses, the probability of getting heads on all 3 tosses is (1/2) * (1/2) * (1/2) = 1/8.
- If this outcome occurs, Mary will win $16.

2. If heads appears on 2 of the tosses:
- The probability of getting heads on a single toss is still 1/2.
- We need to consider the three possible patterns where heads appears on 2 of the tosses: HHT, HTH, or THH.
- The probability of any of these patterns occurring is (1/2) * (1/2) * (1/2) = 1/8.
- If this outcome occurs, Mary will win $2.

3. If heads appears on less than 2 tosses:
- The only remaining outcome is when heads appears on only 1 or none of the tosses.
- The probability of getting tails on a single toss is 1/2.
- The probability of this outcome occurring is 1 - (1/8) - (1/8) = 3/4 (since the sum of probabilities must be 1).
- If this outcome occurs, Mary will lose $5 because she had to pay the cost of playing the game.

Now we can calculate Mary's mean expectation:

Mean expectation = (Probability of the first outcome * Amount won in the first outcome) + (Probability of the second outcome * Amount won in the second outcome) + (Probability of the third outcome * Amount won in the third outcome)

= (1/8 * $16) + (1/8 * $2) + (3/4 * -$5)

= $2 + $0.25 - $3.75

= -$1.50

Therefore, Mary's mean expectation is -$1.50. This means that, on average, she can expect to lose $1.50 for each game she plays.

Assuming Mary gets nothing from the other outcomes.

For 3 tosses,
outcome probability(P) return(E)
HHH 1/8 $16
HHT 1/8 $2
HTH 1/8 $2
THH 1/8 $2
Mean expectation
= Σ PE
= 16/8 + 3*2/8
= 2+6/8
= 2.75