In the game of roulette, a wheel consists of 38 slots numbered 0, 00, 1, 2,..., 36. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you selected, you win $35; otherwise you lose $1.A)Construct a probability distribution for the random variable X, the winnings of each spin.
X P(X)
35 .0263
-1 .9737
(b) Determine the mean and standard deviation of the random variable X. Round your results to the nearest penny.

a) Probability distribution of the winnings is given in the table below

X ||| P(X)

35 | 0.0263

-1 | 0.9737

b) Mean = -$0.0532 = -$0.0532 to the nearest penny; that is, 5 pennies.

Standard deviation = $5.76 = 576 pennies to the nearest penny.

Step-by-step explanation:

To construct the probability distribution, we first define the random variable X which represents the winnings of each spin

X can take on value of $35; that is, a win of $35 when the number of the slot the ball falls into matches the number you selected

And a value of -$1; that is, a loss of $1 if the number of the slot the ball falls into does not match the number you selected.

Note that there are 38 slots. And one can only select 1 slot at a time.

So, the probability that one wins when the slot that one selects is the right slot = (1/38) = 0.0263

Probability of losing = 1 - 0.0263 = 0.9737

So, the probability distribution is now given as

X ||| P(X)

35 | 0.0263

-1 | 0.9737

b) Mean and Standard deviation of the random variable X

Mean = Expected value.

Expected value = E(X) = Σ xᵢpᵢ

where xᵢ = each variable

pᵢ = probability of each variable

E(X) = (35)(0.0263) + (-1)(0.9737)

= 0.9205 - 0.9737 = -$0.0532 = -$0.05 to the nearest penny; thanks 5 pennies to the nearest penny.

Standard deviation = √(variance)

Variance = Var(X) = Σx²p − μ²

where μ = E(X) = -0.0532

Σx²p = (35²)(0.0263) + (-1)²(0.9737)

= 32.2175 + 0.9737 = 33.1912

Var(X) = 33.1912 - (-0.0532)² = 33.18836976

Standard deviation = √(variance)

= √(33.18836976) = $5.76; that is, 576 pennies.

Hope this Helps!!!

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