John is watching an old game show rerun on television called Let’s Make a Deal in...

John is watching an old game show rerun on television called Let’s Make a Deal in which the contestant chooses a prize behind one of two curtains. Behind one of the curtains is a gag prize worth $130, and behind the other is a round-the-world trip worth $8,500. The game show has placed a subliminal message on the curtain containing the gag prize, which makes the probability of choosing the gag prize equal to 75 percent. What is the expected value of the selection? (Round answer to 2 decimal places, e.g. 15.25.) Expected value $ 2222.50 LINK TO TEXT What is the standard deviation of that selection? (Round answer to 2 decimal places, e.g. 15.25.) Standard deviation $

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Expert Solution

Expected Value

The expected value is often referred to as the “long-term” average or mean. This means that over the long term of doing an experiment over and over, you would expect this average.

To find the expected value or long term average, μ, simply multiply each value of the random variable by its probability and add the products.

Mean or Expected Value:
μ=∑ (X.P(x))

In this Case,

Prize (X)	Probability (P(x))	Prize x Probability (X.P(x))
$130	0.75	97.50
$8500	0.25	2125.00
Mean or Expected Value (Total)	2222.50

Standard Deviation:

The standard deviation of a random variable X measures how close the random variable is to the mean (Expected Value). It is called a standard deviation since it represents an “average” (or standard) distance (or deviation) from the mean.

To find the standard deviation σ for a random variable, we

(Compute deviations.) For each value of the random variable X, we compute the difference (or deviation) of X from the mean value.
(Square deviations.) We then square each of the differences that we computed in step 1.
(Compute the average squared deviation.) We find the average squared deviation, by multiplying each squared deviation by the corresponding probability, and summing the products.
(Take square root.) The standard deviation is the square root of the average squared deviation.

Standard Deviation:
σ=√ ∑X(x−μ)² P(x)

In this case,

Prize (X)	P(x)	(X-μ)	(X-μ)²	(X-μ)². P(x)
$130	0.75	-2092.5	4378556.25	3283917.1875
$8500	0.25	6277.5	39407006.25	9851751.5625
Total	13135668.75
Standard Deviation =√ ∑X(x−μ)² P(x)	3624.32

jeff jeffy answered 1 year ago

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