Question

Three coins are tossed. Let the event H = all Heads and the event K = at least one Heads. (match like 1-a, etc)

Answer 1

Given:

Three coins are tossed.

Let the event H represents all Heads and the event K represents at least one Heads.

To find:

a. The probability that the outcome is all heads if at least one coin shows a heads.

b. P(K) = ?

c. P(H∩K) = ?

Solution:

If three coins are tossed, then the total possible outcomes are:

{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Total outcomes = 8

Possible outcomes for all Heads = 1

Possible outcomes for at least one Heads = 7

Let the following events:

H = all Heads

K = at least one Heads.

Then,

$H=\{HHH\}$

$K=\{HHH, HHT, HTH, HTT, THH, THT, TTH\}$

$H\cap K=\{HHH\}$

Now,

$P(K)=\dfrac{7}{8}$

$P(H\cap K)=\dfrac{1}{8}$

a. The probability that the outcome is all heads if at least one coin shows a heads is:

$P(H|K)=\dfrac{P(H\cap K)}{P(K)}$

$P(H|K)=\dfrac{\dfrac{1}{8}}{\dfrac{7}{8}}$

$P(H|K)=\dfrac{1}{7}$

Therefore, the probability that the outcome is all heads if at least one coin shows a heads is $\dfrac{1}{7}$.

b. $P(K)=\dfrac{7}{8}$

c. $P(H\cap K)=\dfrac{1}{8}$