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Chapter 6: Problem 7

The following table lists the probability distribution for cash prizes in a lottery conducted at Lawson's Department Store. $$\begin{array}{|rc|}\hline \text { Prize (\$) } & \text { Probability } \\\\\hline 0 & .45 \\\10 & .30 \\\100 & .20 \\\500 & .05 \\\\\hline\end{array}$$ If you buy a single ticket, what is the probability that you win: a. Exactly \(\$ 100 ?\) b. At least \(\$ 10 ?\) c. No more than \(\$ 100 ?\) d. Compute the mean, variance, and standard deviation of this distribution.

Short Answer

Expert verified
a. 0.20 b. 0.55 c. 0.95 d. Mean = $48, Variance = 10226, SD 鈮� 101.12

Step by step solution

01

Determine Probability for Exactly $100

The probability of winning exactly \(100 is given in the table. Simply locate the value for the probability corresponding to \)100. The probability is \( P(\text{Win } \$100) = 0.20 \).
02

Calculate Probability for at least $10

To find the probability of winning at least \(10, sum the probabilities of winning \)10, \(100, and \)500. Thus: \( P(\text{Win at least } \$10) = 0.30 + 0.20 + 0.05 = 0.55 \).
03

Calculate Probability for No More than $100

To find the probability of winning no more than \(100, sum the probabilities of winning \)0, \(10, and \)100. Thus: \( P(\text{Win no more than } \$100) = 0.45 + 0.30 + 0.20 = 0.95 \).
04

Compute the Mean of the Distribution

The mean is calculated by multiplying each prize amount by its probability and summing the results: \( \text{Mean} = 0 \times 0.45 + 10 \times 0.30 + 100 \times 0.20 + 500 \times 0.05 = 0 + 3 + 20 + 25 = 48 \).
05

Compute the Variance of the Distribution

The variance is calculated by finding the expected value of the square of each outcome minus the square of the mean: \( \text{Variance} = (0^2 \times 0.45 + 10^2 \times 0.30 + 100^2 \times 0.20 + 500^2 \times 0.05) - 48^2 = (0 + 30 + 2000 + 12500) - 2304 = 10226 \).
06

Calculate the Standard Deviation

The standard deviation is the square root of the variance: \( \text{Standard Deviation} = \sqrt{10226} \approx 101.12 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
The calculation of the mean provides us with an average prize that you might expect to win per ticket purchase in a lottery. To determine the mean of this probability distribution, you need to perform a weighted average calculation.
This involves multiplying each prize by its probability and adding these products together.
For example, the prize of \(10 with a probability of 0.30 contributes to the mean as follows:
  • Multiply: \( 10 \times 0.30 = 3 \)
This process is repeated for all possible prizes, including:
  • \( 0 \times 0.45 = 0 \)
    \( 100 \times 0.20 = 20 \)
    \( 500 \times 0.05 = 25 \)
Adding these values gives you the mean: \[0 + 3 + 20 + 25 = 48\]This mean value tells us that, on average, the amount won per ticket purchased is \)48.
Variance Calculation
Variance provides insight into the amount of spread in your probability distribution.
It's useful to understand how much variation or "risk" you have in the prizes.
To calculate variance, we must first determine the expected value of the squares of the outcomes and then subtract the square of the mean from this value.
  • Calculate the squared prize times its probability:
    \(0^2 \times 0.45 = 0\)
    \(10^2 \times 0.30 = 30\)
    \(100^2 \times 0.20 = 2000\)
    \(500^2 \times 0.05 = 12500\)
Add these results: \[0 + 30 + 2000 + 12500 = 14530\]Remember the mean from before: \(48\). Now, square the mean:
  • \(48^2 = 2304\)
Finally, subtract this from the sum of the squared outcomes:\[14530 - 2304 = 12226\]Thus, the variance is \(12226\). This higher value suggests significant variation in the outcomes.
Standard Deviation
The standard deviation is a measure of how the prize amounts vary relative to the mean prize.
It's crucial for interpreting the dispersion or spread of the data.
Since variance is in squared units, the standard deviation brings the value back down to the original units (dollars in this case) by taking the square root of the variance.
  • Take the square root of the calculated variance:
    \(\text{Standard Deviation} = \sqrt{12226} \approx 110.57\)
The resulting standard deviation of approximately \(110.57\) tells us that the cash prizes tend to differ from the mean by about this amount on average.
It gives a clearer view of the average variability in the prizes relative to the mean.

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