Question

Exercise 12.1.1: Coin flips and events. About A coin is flipped four times. For each of the events described below, express the event as a set in roster notation. Each outcome is written as a string of length 4 from {H, T}, such as HHTH. Assuming the coin is a fair coin, give the probability of each event. (a) The first and last flips come up heads. (b) There are at least two consecutive flips that come up heads. (c) The first flip comes up tails and there are at least two consecutive flips that come up heads.

Answer 1

a
`Pr = 0.25`

b
`Pr = 0.50`

c
`Pr = 0.1875`

Explanation:

Given

`n = 4` --- number of times

`r = 2` --- faces of a coin

First, is to determine the sample size.

This is calculated as:

`Size =r^n`

`Size =2^4`

`Size =16`

Solving (a): First and last slip is head

This event is represented as:

`E=\{HHHH, HHTH, HTHH, HTTH\}`

The probability is calculated as:

`Pr = (n(E))/(Size)`

`Pr = (4)/(16)`

`Pr = 0.25`

b) At least 2 consecutive flips that is heads.

This event is represented as:

`E=\{HHHH,HHHT,HHTH,HHTT,THHH,THHT,HTHH,TTHH\}`

The probability is calculated as:

`Pr = (n(E))/(Size)`

`Pr = (8)/(16)`

`Pr = 0.50`

c) First is tail and at least 2 consecutive flips is head.

This event is represented as:

`E=\{THHH,THHT,TTHH\}`

The probability is calculated as:

`Pr = (n(E))/(Size)`

`Pr = (3)/(16)`

`Pr = 0.1875`