Question

Exercise 5.16.1: 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. Solution (b) There are at least two consecutive flips that come up heads.

Answer 1

`(a)\ Pr = 0.25`

`(b)\ Pr = 0.50`

Explanation:

Given

`Flips = 4`

First, we calculate the sample size;

The coin has two possible outcomes (i.e. head or tail) and the number of flips is 4.

So, the sample size (N) is:

`N = 2^4`

`N = 16`

Solving (a): Probability the outcome of the first and last flip is head.

The possible outcomes for this event are:

`n = \{(HHHH), (HHTH), (HTHH),(HTTH)\}`

`n(n) = 4`

The probability is:

`Pr = (n(n))/(N)`

`Pr = (4)/(16)`

`Pr = 0.25`

Solving (a): Probability that the outcome of at least 2 consecutive flips is head.

The possible outcomes for this event are:

`n = \{(HHHH), (HHHT),(HHTH), (HTHH),(THHH),(TTHH),(HHTT), (THHT)\}`

`n(n) = 8`

The probability is:

`Pr = (n(n))/(N)`

`Pr = (8)/(16)`

`Pr = 0.50`