Question

28% of males smoke cigarettes, 7% smokes cigars, 3% smoke both.

if a male smokes, what is the probability that he smokes only cigars not cigarettes?  Solve analytically: first write the probability statement in terms of sets A and B , union/intersection operators, etc., then compute the probability.

Answer 1

Here let say event

A = smoking cigarettes

B = smoke cigars

so here as in the questioon

Pr(A) = 28% = 0.28

Pr(B) = 7% = 0.07

Pr(A ∩ B) = 3% = 0.03

Now as the question given here that is a male smokes, we have to find that he smokes only cigars not cigrattes.

writing the probability statement we have to find is the probability that a male smokes

Pr(A male smokes) = Pr(A U B) = Pr(A) + Pr(B) - Pr(A ∩ B) = 0.28 + 0.07 - 0.03 = 0.32

so now,

Pr(only cigars not cigarettes) = Pr(cigars) - Pr(Cigars and cigerettes both) = Pr(B) - Pr(A ∩ B) = 0.07 - 0.03 = 0.04

so her,

Pr(Only smokescigars not cigarettes when a person smokes) = 0.04/0.32 = 0.125

Answer 2

Let's define the following events: A: The event that a male smokes cigarettes. B: The event that a male smokes cigars.

We are given the following probabilities: P(A) = 0.28 (probability that a male smokes cigarettes) P(B) = 0.07 (probability that a male smokes cigars) P(A ∩ B) = 0.03 (probability that a male smokes both cigarettes and cigars)

We want to find the probability that a male smokes only cigars, not cigarettes, i.e., P(B - A).

The probability statement in terms of sets A and B is: P(B - A) = P(B ∩ A') / P(A)

where A' represents the complement of A (the event that a male does not smoke cigarettes).

Now, let's compute the probability:

1. First, find P(A'): The probability that a male does not smoke cigarettes. P(A') = 1 - P(A) = 1 - 0.28 = 0.72

2. Next, find P(B ∩ A'): The probability that a male smokes cigars but not cigarettes. P(B ∩ A') = P(B) - P(A ∩ B) = 0.07 - 0.03 = 0.04

3. Finally, find P(B - A): The probability that a male smokes only cigars, not cigarettes. P(B - A) = P(B ∩ A') / P(A) = 0.04 / 0.28 ≈ 0.143

So, the probability that a male smokes only cigars, not cigarettes, is approximately 0.143 or 14.3%.

Answer 3

Let's define the following events:

A: Male smokes cigarettes. B: Male smokes cigars.

We are given the following probabilities:

P(A) = 0.28 (probability of a male smoking cigarettes) P(B) = 0.07 (probability of a male smoking cigars) P(A ∩ B) = 0.03 (probability of a male smoking both cigarettes and cigars)

We want to find the probability that a male smokes only cigars (not cigarettes), which can be denoted as P(B - A).

Using set notation and probabilities, the probability statement can be written as:

P(B - A) = P(B ∩ A') / P(A)

where A' represents the complement of A, i.e., the event that a male does not smoke cigarettes.

Now, let's compute the probability:

1. Calculate P(A'):

P(A') = 1 - P(A) P(A') = 1 - 0.28 P(A') = 0.72

2. Calculate P(B ∩ A'):

P(B ∩ A') = P(B) - P(A ∩ B) P(B ∩ A') = 0.07 - 0.03 P(B ∩ A') = 0.04

3. Calculate P(B - A):

P(B - A) = P(B ∩ A') / P(A) P(B - A) = 0.04 / 0.28

Now, divide 0.04 by 0.28:

P(B - A) ≈ 0.142857

So, the probability that a male smokes only cigars (not cigarettes) is approximately 0.142857 or 14.29%.