Answer:
$19.44
Step-by-step explanation:
Solution:-
- The hat contains red, green and yellow balls.
- The colored balls are distributed such that there are:
Color Number of balls
Red Balls ( R ) 3
Green Balls ( G ) 5
Yellow Balls ( Y ) 1
Total 9
- The amount that we either win or loose when a ball is drawn from the hat is also distributed according to color. Positive number are shown as gain and negative numbers are given as loss.
Color Win/Loss ( $ )
Red Balls ( R ) 4
Green Balls ( G ) -3
Yellow Balls ( Y ) 10
- We are to play the game for n = 25 times.
- The probability of selecting any ball from the hat remains constant under the condition of ( replacement ). Meaning that any ball drawn is placed back into the hat.
- The probability of selecting any colored ball is independent from selecting other colored ball. Moreover, the probability of selecting the same colored ball is also independent across every successive trial/draw.
- The probabilities for each colored ball are determined from the basic definition ( Favorable outcomes / Total outcomes ).
Color Probability ( p = number of balls / Total )
Red Balls ( R ) 3 / 9
Green Balls ( G ) 5 / 9
Yellow Balls ( Y ) 1 / 9
- We will define 3 random variables as follows:
X: The number of red balls drawn from hat
Y: The number of Green balls drawn from hat
Z: The number of Yellow balls drawn from the hat
Note: Each random variable follows "Binomial distribution" under the conditions applied above ( Replacement and independent events ).
- The expected number of colored balls drawn in n = 25 trials/draws.
Color Expected number = n*p
Red Balls ( R ) E ( X ) = 25*(3/9) = 8.33
Green Balls ( G ) E ( Y ) = 25*(5/9) = 13.889
Yellow Balls ( Y ) E ( Z ) = 25*(1/9) = 2.7778
- We will define a random variable W: The amount of gain/loss from the game.
- The expected value of a random variable ( W ) i.e the expected gain/loss from playing the game n = 25 times is defined as:
E ( W ) = Sum ( expected number*win/loss )
Color Expected number = n*p Win/Loss ( $ ) Expected Gain
Red Balls ( R ) E ( X ) = 25*(3/9) = 8.33 4 8.33*4 = $300/9
Green Balls ( G ) E ( Y ) = 25*(5/9) = 13.888 -3 13.8*-3 = -$375/9
Yellow Balls ( Y ) E ( Z ) = 25*(1/9) = 2.7778 10 10*2.8 = $250/9
Where,
E ( W ) = Sum of last column
= 300/9 - 375/9 + 250/9
= 175 / 9 = $19.44444
Answer: There is an expected gain of $19.44 from playing the game 25 times.
