Question 5 OR (d)

Selling at least one of two products, and both products if the sales were independent

4 marks

Time to write
About 8 minutes
Topic
Probability and Bayes' theorem

Question

Question 5 OR (d) · 4 marks

On any given day a sales representative can sell Product A with probability 0.5, and the same representative can sell Product B on that day with probability 0.4. The probability that the representative sells both products on the same day is 0.15. What is the probability that the representative sells at least one of the two products on any given day? If selling Product A and selling Product B are taken to be independent events, what is the probability that the representative sells both products on the same day?

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

8 minfor 4 marks

Read from the question

  • What is the…?Are…?Two asks in one question: answer both.

What the paper leaves open

As printed, the stated probability of selling both products is not the product of the two single-product probabilities (it is smaller); the question then asks again for both products under independence. Both parts are answered as asked.

Write it in this order

  1. Read the data; note exactly what is asked 1 min
  2. 1 Addition theorem and the probability of at least one sale 2 marks 3 min
  3. 2 Probability of both sales under independence 1½ marks 2 min
  4. 3 Comment on the stated joint probability ½ mark 2 min

Short on time? Step 1 carries 2 of the 4 marks: guard its minutes.

Try it first. About 8 minutes for 4 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Events. Let A be the event that the representative sells Product A on a given day and B the event of selling Product B: P(A) = 0.5, P(B) = 0.4 and, as given, P(A ∩ B) = 0.15.

At least one product sold

By the addition theorem for any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B):

P(A ∪ B) = 0.5 + 0.4 − 0.15 = 0.75.

Outcome on a dayWorkingProbability
Both productsgiven0.15
Product A only0.5 − 0.150.35
Product B only0.4 − 0.150.25
At least one product0.75

On three days out of four the representative sells at least one of the products.

Both products sold, if the events are independent

For independent events, P(A ∩ B) = P(A) × P(B):

P(A ∩ B) = 0.5 × 0.4 = 0.2.

Comment on the data. The joint probability stated in the question (0.15) is smaller than the product 0.2, so on the question's own figures the two sales are not independent: selling one product makes the other slightly less likely (P(B | A) = 0.15 / 0.5 = 0.3, below P(B) = 0.4), as when time spent closing one sale leaves less time for the other. The second part is therefore a hypothetical case, as the paper note says. Under independence the chance of selling at least one product would be 0.5 + 0.4 − 0.2 = 0.7, against 0.75 on the stated data.

Where the 4 marks sit

#ComponentMarks
1Addition theorem and the probability of at least one sale2
2Probability of both sales under independence1.5
3Comment on the stated joint probability0.5
Total4
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