Question

Talia wants to play a basketball game at a carnival. the game cost the player $5 to play, and the player gets to take too long distance shots. if they missed both shots, they get nothing. if they make one shot, they get their $5 back. Thalia has a 40% chance of making this type of shot. here is the probability distribution of x= the amount of money Talia gains from playing the game.

x= money gain -$5 $0 $5
P(x) 0.36 0.48 0.16


Given that μX​=−$1. Calculate σX​
σX​=−−−−−−− dollars.

Answer 1

To calculate the standard deviation of Talia's gains from playing the basketball game, we used the given probabilities and the mean to find the variance first and then took its square root, arriving at approximately $3.46.

Step-by-step explanation:

Talia wants to play a basketball game at a carnival which costs $5 to play. Given the probability distribution of Talia's gain (X) from playing, we can calculate the standard deviation (σX). Talia can either lose $5, break even, or gain $5, with probabilities of 0.36, 0.48, and 0.16 respectively. We also know that the mean (μX) is $-1.

First, we calculate the expected value of X:

Σ xP(x) = (-$5)(0.36) + ($0)(0.48) + ($5)(0.16) = - $1.80 + $0 + $0.80 = - $1

Now, to find the standard deviation, we will calculate the variance (σX2):

Σ [x - E(X)]2P(x) = [(-$5 - (-$1))2(0.36)] + [($0 - (-$1))2(0.48)] + [($5 - (-$1))2(0.16)] = [(-$4)2(0.36)] + [($1)2(0.48)] + [($6)2(0.16)] = ($16)(0.36) + ($1)(0.48) + ($36)(0.16) = $5.76 + $0.48 + $5.76 = $12.00

The standard deviation is the square root of the variance:

σX = √σX2 = √$12.00 ≈ $3.46

Therefore, Talia's standard deviation of money gained from playing the basketball game is approximately $3.46.