A game consists of tossing two coins, first a dime and then a nickel. For each coin which comes up a head you win the value of the coin. For each coin which comes up a tail you get nothing.
{HH,HT,TH,TT} - each outcome with probability .25 - winnings are 15,10,5,0
a) Find the mean or expected winnings
b) Find the standard deviation of your winnings.
c) If you had to pay a dime to play the game, what would be the mean of your winnings?
d) If you had to pay a dime to play the game, what would be the standard deviation of your winnings?
Expected winning can be found by summing over all possible outcomes the product of winning and the respective probability.
For outcome HH:
winning=0.25
probability = 0.25
prodcut = 0.25*0.25=0.0625
Repeat for the 3 other outcomes, and add the 4 products together to get the expected winning.
The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 170 and a variance of 4. The material is considered defective if the breaking strength is less than 166 pounds. What is the probability that a single, randomly selected piece of material will be defective?
To find the mean or expected winnings, we need to calculate the sum of the products of each outcome and its corresponding probability. Let's proceed with the calculations:
a) Calculate the mean or expected winnings:
| Outcome | Probability | Winnings |
|---------|-------------|----------|
| HH | 0.25 | 15 |
| HT | 0.25 | 10 |
| TH | 0.25 | 5 |
| TT | 0.25 | 0 |
Expected winnings = (0.25 * 15) + (0.25 * 10) + (0.25 * 5) + (0.25 * 0) = 3.75 + 2.5 + 1.25 + 0 = 7.5
Therefore, the mean or expected winnings are $7.50.
b) To find the standard deviation of your winnings, we need to calculate the variance first. Then, take the square root of the variance to find the standard deviation. Let's proceed with the calculations:
| Outcome | Probability | Winnings | Difference (Winnings - Expected Winnings) | Squared Difference |
|---------|-------------|----------|------------------------------------------|--------------------|
| HH | 0.25 | 15 | 7.5 | 56.25 |
| HT | 0.25 | 10 | 2.5 | 6.25 |
| TH | 0.25 | 5 | -2.5 | 6.25 |
| TT | 0.25 | 0 | -7.5 | 56.25 |
Variance = (0.25 * 56.25) + (0.25 * 6.25) + (0.25 * 6.25) + (0.25 * 56.25) = 14.0625 + 1.5625 + 1.5625 + 14.0625 = 31.25
Standard Deviation = √Variance = √31.25 = 5.59 (rounded to two decimal places)
Therefore, the standard deviation of your winnings is approximately $5.59.
c) If you had to pay a dime to play the game, the mean of your winnings would decrease by 10 cents. So, the new mean would be $7.50 - $0.10 = $7.40.
Therefore, if you had to pay a dime to play the game, the mean of your winnings would be $7.40.
d) The standard deviation does not change when you subtract a constant value from each outcome. So, if you had to pay a dime to play the game, the standard deviation of your winnings would still be approximately $5.59.
To find the mean or expected winnings, we need to calculate the weighted average of the winnings for each possible outcome.
a) Mean or Expected Winnings:
To calculate the expected winnings, we multiply each winning amount by its corresponding probability and then sum them up.
Probabilities for each outcome:
P(HH) = 0.25
P(HT) = 0.25
P(TH) = 0.25
P(TT) = 0.25
Winnings for each outcome:
W(HH) = $15
W(HT) = $10
W(TH) = $5
W(TT) = $0
Expected Winnings = (P(HH) * W(HH)) + (P(HT) * W(HT)) + (P(TH) * W(TH)) + (P(TT) * W(TT))
Expected Winnings = (0.25 * $15) + (0.25 * $10) + (0.25 * $5) + (0.25 * $0)
Expected Winnings = $3.75 + $2.50 + $1.25 + $0
Expected Winnings = $7.50
Therefore, the mean or expected winnings for this game are $7.50.
b) Standard Deviation of Winnings:
To calculate the standard deviation, we need to find the variance first. The variance is the average of the squared differences from the mean.
Variance = [ (W(HH) - Mean)^2 * P(HH) ] + [ (W(HT) - Mean)^2 * P(HT) ] + [ (W(TH) - Mean)^2 * P(TH) ] + [ (W(TT) - Mean)^2 * P(TT) ]
Variance = [ ($15 - $7.50)^2 * 0.25 ] + [ ($10 - $7.50)^2 * 0.25 ] + [ ($5 - $7.50)^2 * 0.25 ] + [ ($0 - $7.50)^2 * 0.25 ]
Variance = [ $7.50^2 * 0.25 ] + [ $2.50^2 * 0.25 ] + [ (-$2.50)^2 * 0.25 ] + [ (-$7.50)^2 * 0.25 ]
Variance = $1.875 + $0.3125 + $0.3125 + $1.875
Variance = $4.375
Standard Deviation = √Variance = √$4.375
Standard Deviation ≈ $2.09
Therefore, the standard deviation of the winnings is approximately $2.09.
c) Mean of Winnings with a $0.10 fee:
If you had to pay $0.10 to play the game, you should subtract this fee from the expected winnings.
Mean of Winnings with fee = Expected Winnings - Fee
Mean of Winnings with fee = $7.50 - $0.10
Mean of Winnings with fee = $7.40
Therefore, if you had to pay a dime to play the game, the mean of your winnings would be $7.40.
d) Standard Deviation of Winnings with a $0.10 fee:
The standard deviation will remain the same because adding or subtracting a constant value from each outcome does not affect the spread of the data.
Therefore, the standard deviation of the winnings with a $0.10 fee would still be approximately $2.09.