Question

3. Binomial Random Variable: On your way home from work at 9pm every day You estimate that 25% of the time, they have at least one slice of pepperoni pizza (your favorite) from work at 9pm every day, you pass your favorite pizza store. remaining, 75% of the time, they have already sold out of slices. Assume that whenever there is pizza available, you will buy one slice. Consider the last 10 days and let X represent the number of pizza slices you purchased on the way home from work. a. For this experiment, define the event that represents a "success" b. Explain why X is (approximately) a binomial random variable. c. Give the value of p for this binomial experiment. d. Find P(x<5). e. Find P(x-3). f. Find P(x>o). g How many slices of pepperoni pizza do you expect to purchase over a time period of 10 days?

Answer 1

a) As X = the number of slices purchased in the last 10 days,

The event that represents a success = Pizza is available on a given day i.e you're able to buy a slice of pepperoni pizza on a given day.

b) X is a binomial random variable because :

i) X has a fixed probability of success for each trial.

ii) There are only two possible outcomes for each trial.

iii) Each of the 10 trials are independent with each other.

c) The value of p (the value of success for a trial) = 0.25

d) X can take the values = [0:10]

P(X<5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$

$P(X=0) = \binom{10}{0} (0.25))^0 (0.75)^{10}$

$P(X=1) = \binom{10}{1} (0.25)^1 (0.75)^{9}$

$P(X=2) = \binom{10}{2} (0.25))^2 (0.75)^{8}$

$P(X=3) = \binom{10}{3} (0.25))^3 (0.75)^{7}$

$P(X=4) = \binom{10}{4} (0.25))^4 (0.75)^{6}$

P(X < 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)

P(X < 5) = (0.75)¹⁰ + [10(0.25)¹(0.75)⁹] + [45(0.25)²(0.75)⁸] + [120(0.25)³(0.75)⁷] + [ 210(0.25)⁴(0.75)⁶]

P(X < 5) = (3/4)¹⁰ + 10(3)⁹ / (4)¹⁰ + 45(3)⁸ / (4)¹⁰ + 120(3)⁷ / (4)¹⁰ + 210(3)⁶ / (4)¹⁰  

P(X < 5) = (3)⁶/(4)¹⁰ [ (3)⁴ + 10(3)³ + 45(3)² + 120(3)¹ + 210]

P(X < 5) = 729 / 1048576 [ 81 + 270 + 405 + 360 + 210]

P(X < 5) = 966,654 / 1048576

Answer : P(X < 5) = 0.922

e)

$P(X=3) = \binom{10}{3} (0.25))^3 (0.75)^{7}$

P(X = 3) = 120 x 3⁷ / 4¹⁰

P(X = 3) = 0.25

f) P(X > 0) = 1 - P(X <= 0)

Since X can only be 0 ->

P(X > 0) = 1 - P(X = 0)

$P(X=0) = \binom{10}{0} (0.25))^0 (0.75)^{10}$

P( X = 0 ) = 0.056

P( X > 0 ) = 1 - 0.056

P(X > 0) = 0.944

g) Expected value of X = E(X)

E(X) for a Binomial Random Variable (X) = np

Mean = np

Where n = 10 and p = 0.25

Therefore, E(X) = 10 x 0.25 = 2.5

Ans: Mean slices of pizza bought over 10 days = 2.5 slices

Cheers!

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