Question

You flip a coin, if it is heads you will have a good day and if it is tails you will have a bad day. There are 30 days in total.

(a) What is the expectation and variance of the number of times you will have a good day throughout this 30 day stretch?

(b) What is the probability that every day will be bad for all of the 30 days?

Answer 1

Solution:

Given: A coin is flipped everyday , throughout 30 days.

If it is heads you will have a good day and if it is tails you will have a bad day.

Let X = Number of good days out of 30 days

Assume coin is fair , then p = probability of good day = 0.5 and q = probability of bad day = 0.5

n = Total number of days = 30

Thus X = Number of good days follows a Binomial distribution with parameters n = 30 and p = 0.5

Part a) What is the expectation and variance of the number of times you will have a good day throughout this 30 day stretch?

For Binomial distribution , the expectation or expected value of X = number of times you will have a good day is given by:

and variance of the number of times you will have a good day:

Part b)

What is the probability that every day will be bad for all of the 30 days?

It says all 30 days would be Bad, then X = Number of Good Days = 0

That is we have to find:

P(X = 0 ) = ........?

Using Binomial probability :

where

Thus

or

or rounded to 4 decimal places.

Thus there is approximately 0 probability that every day will be bad for all of the 30 days.