Vera examined a random sample of deodorants distributed by her company. She found t...
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Problem
Vera examined a random sample of deodorants distributed by her company. She found the following probabilities: P(for men) = 0.55, P(botanic scent) = 0.18, P(for men and botanic) = 0.05. Find the probability that a deodorant is for men, given that it has a botanic scent. P (for men | botanic) = ?
Answer
\(\frac{5}{18} \approx 0.278\)
Step-by-step solution
1. Identify the given probabilities:
We are given the following probabilities:
- The probability that a deodorant is for men: $P(\text{for men}) = 0.55$
- The probability that a deodorant has a botanic scent: $P(\text{botanic scent}) = 0.18$
- The probability that a deodorant is for men AND has a botanic scent: $P(\text{for men and botanic}) = 0.05$
2. Understand the question:
We need to find the conditional probability that a deodorant is for men, GIVEN that it has a botanic scent. This is written as $P(\text{for men} | \text{botanic scent})$.
3. Recall the formula for conditional probability:
The formula for conditional probability is:
$$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$$
In this case, let A be the event "for men" and B be the event "botanic scent".
4. Apply the formula:
Substitute the given values into the formula:
$$P(\text{for men} | \text{botanic scent}) = \frac{P(\text{for men and botanic scent})}{P(\text{botanic scent})}$$
$$P(\text{for men} | \text{botanic scent}) = \frac{0.05}{0.18}$$
5. Calculate the result:
Divide the numbers:
$$P(\text{for men} | \text{botanic scent}) = \frac{0.05}{0.18} = \frac{5}{18}$$
To express this as a decimal, we perform the division:
$$5 \div 18 \approx 0.277777...$$