There are 4 people in a car. One person is wearing glasses. 2 people are wearing hats. Explain why the probability of a person in the car wearing glasses or a hat is not necessarily 3/4

Relevance
• 4 weeks ago

P(wear glasses) = 1/4

P(wear no glasses) = 1 - (1/4) = 3/4

P(wear a hat) = 2/4 = 1/2

P(wear no hat) = 1 - (1/2) = 1/2

====

Method 1:

P(wear glasses or a hat)

= 1 - P(wear no glasses and no hot)

= 1 - (3/4) × (1/2)

= 1 - (3/8)

= 5/8

====

Method 2:

P(wear glass or a hat)

= P(wear glasses but no hat) + P(wear no glasses but a hat) + P(wear glasses and a hat)

= (1/4) × (1/2) + (3/4) × (1/2) + (1/4) × (1/2)

= (1/8) + (3/8) + (1/8)

= 5/8

• atsuo
Lv 6
4 weeks ago

Given condition : There are 4 people in a car. One person is

wearing glasses. 2 people are wearing hats.

The condition shows us that there are two possibilities.

(1) 1 person wears glasses, other 2 persons wear a hat.

So 1 person does not wear glasses nor a hat.

(2) 1 person wears glasses and a hat, another person wears a hat.

So 2 persons do not wear glasses nor a hat.

In (1), 3 persons wear glasses or a hat so the answer becomes 3/4.

In (2), 2 persons wear glasses or a hat so the answer becomes 1/2.

But given condition does not tell us which of (1) and (2) occurs.

If we know that the probability that (1) occurs is p then the answer

becomes (3/4)p+(1/2)(1-p). But p is unknown , so I think the answer

becomes "at least 1/2, at most 3/4" .

• A.J.
Lv 7
4 weeks ago

We don't know whether the person wearing glasses ALSO has a hat on. More so, to get fancy about this, we don't know whether it's sunglasses and the person with glasses is more likely to also wear a hat for sun shading. We don't know whether the driver must wear glasses or restricted from a hat when driving.

It could be random, related, restricted, or exclusive as one or the other.

You can have one with glasses and hat and one with hat only.

You can have one with glasses and two with hat only.

The actual probabilities are a later class.