People Who Can Wiggle Their Ears In a sample of 1000 people

People Who Can Wiggle Their Ears In a sample of 1000 people, 130 can wiggle their ears. Two unrelated people are selected at random without replacement.

(a) Find the probability that both people can wiggle their ears.

(b) Find the probability that neither person can wiggle his or her ears.

(d) Which of the events can be considered unusual? Explain.

The correct answer and explanation is :

Alright, let’s walk through this carefully:

Given:

Total people = 1000
People who can wiggle their ears = 130
People who cannot wiggle their ears = 1000 – 130 = 870

Two people are selected without replacement.

(a) Probability that both can wiggle their ears:

First person: probability = 130/1000
Second person: probability = 129/999 (since one ear-wiggler is already taken)

Multiply them:
[
P(\text{both wiggle}) = \frac{130}{1000} \times \frac{129}{999}
]
[
= \frac{130 \times 129}{1000 \times 999}
]
[
= \frac{16770}{999000}
]
[
\approx 0.01678
]

✅ Answer: about 0.0168

(b) Probability that neither can wiggle their ears:

First person: probability = 870/1000
Second person: probability = 869/999

Multiply them:
[
P(\text{neither wiggle}) = \frac{870}{1000} \times \frac{869}{999}
]
[
= \frac{870 \times 869}{1000 \times 999}
]
[
= \frac{756030}{999000}
]
[
\approx 0.75679
]

✅ Answer: about 0.7568

(c) Probability that at least one can wiggle:

Use complement rule:
[
P(\text{at least one wiggle}) = 1 – P(\text{neither wiggle})
]
[
= 1 – 0.7568
]
[
= 0.2432
]

✅ Answer: about 0.2432

(d) Which event is unusual?

An event is often considered unusual if its probability is less than 0.05.

(a) Probability of both wiggling = 0.0168 → Yes, unusual (since 0.0168 < 0.05)
(b) Probability of neither wiggling = 0.7568 → No, not unusual
(c) Probability of at least one wiggling = 0.2432 → No, not unusual

✅ Answer: Event (a) (“both wiggle”) is unusual.

➔ 300-word explanation:

In this problem, we are analyzing the probability of people who can wiggle their ears. Out of a sample of 1000 people, 130 individuals possess this unique ability. When two people are randomly selected without replacement, the probability that both individuals can wiggle their ears is calculated by multiplying the individual probabilities. For the first person, there are 130 successful outcomes out of 1000, and after one person is chosen, 129 successful outcomes remain out of 999 total. This yields a probability of about 0.0168 (or 1.68%), which is relatively low.

Next, we consider the probability that neither person can wiggle their ears. Since 870 people cannot wiggle their ears, the probability for the first person is 870 out of 1000, and then 869 out of 999 for the second. Multiplying these gives a probability of about 0.7568 (or 75.68%), showing it is very common for two randomly selected people to both lack this ability.

The probability that at least one can wiggle their ears is found using the complement rule. Since “at least one” is the opposite of “neither,” we subtract the probability of neither from 1. This gives about 0.2432 (or 24.32%), meaning it’s somewhat likely but not overwhelming.

Finally, we consider which event is unusual. In general, probabilities below 0.05 (5%) are considered unusual. Only the event where both people can wiggle their ears falls below this threshold (0.0168 < 0.05). Thus, it would be unusual for two randomly selected people to both be able to wiggle their ears.