Meeting at least one​ left-handed person in six random encounters on campus when the incidence rate is 18​% ​(18 in 100 people are​ left-handed)

Answer 1

There are two ways of solving this problem. Let's go with the easier one.

Note that there are several ways you could have at least one person be left-handed out of 6 people: 1, 2, 3, 4, 5, or all 6 people are left handed. However let's look at the opposite. How many ways are there for nobody to be left-handed? Just one way: every person is right-handed.

So, we calculate the probability of getting 6 random people that are all right-handed.

*Note that each person is independent of the next, so we multiply the probabilities.

P(1 random person is right-handed)

= 100% - P(1 random person is left-handed)

= 100% - 18%

= 82%

P(6 random people are all right-handed)

= P(1 random person is right-handed)^6

= (82%)^6

= 0.3040066714

So, this situation is the exact opposite of what we are looking for. Thus:

P(at least 1 person out of 6 random people is left-handed)

= 100% - P(6 random people are all right-handed)

= 1 - 0.3040066714

= 0.6959933286

So, the probability of at least 1 person among 6 randomly chosen people is left-handed is about 69.6%