The Mathematics of Maybe: Introduction to Probability | Exercise 7.4

Question 1

There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket.

(i) Draw a tree diagram showing all possible pairs of fruits.

(ii) List the sample space.

(iii) What is the probability of picking one apple and one banana?

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Solution
Understand the Question
  • Basket A has an apple and oranges; Basket B has a banana and a mango.
  • When picking one fruit from each basket, a tree diagram visualizes every combination of choices from Basket A followed by Basket B.
  • The sample space SS is the set of all possible pairs of fruits.
  • Probability is calculated as: P(E)=Number of favourable outcomesTotal number of outcomesP(E) = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}

(i) Draw a tree diagram showing all possible pairs of fruits.

Step 1 · Draw the Tree Diagram

Fruits in Basket A: Apple\text{Apple}, Orange\text{Orange} Fruits in Basket B: Banana\text{Banana}, Mango\text{Mango}Diagram 1

The possible pairs of fruits are:

  • (Apple, Banana)(\text{Apple, Banana})
  • (Apple, Mango)(\text{Apple, Mango})
  • (Orange, Banana)(\text{Orange, Banana})
  • (Orange, Mango)(\text{Orange, Mango})
Answer

(i) Possible pairs: (Apple, Banana)(\text{Apple, Banana}), (Apple, Mango)(\text{Apple, Mango}), (Orange, Banana)(\text{Orange, Banana}), (Orange, Mango)(\text{Orange, Mango})

(ii) List the sample space.

Step 1 · List the Outcomes in Sample Space

The sample space SS is the set of all possible outcomes: S={(Apple, Banana),(Apple, Mango),(Orange, Banana),(Orange, Mango)}S = \{(\text{Apple, Banana}), (\text{Apple, Mango}), (\text{Orange, Banana}), (\text{Orange, Mango})\}

Total number of outcomes =4= 4

Answer

(ii) S={(Apple, Banana),(Apple, Mango),(Orange, Banana),(Orange, Mango)}S = \{(\text{Apple, Banana}), (\text{Apple, Mango}), (\text{Orange, Banana}), (\text{Orange, Mango})\}

(iii) What is the probability of picking one apple and one banana?

Step 1 · Calculate the Probability

Favourable outcome =(Apple, Banana)= (\text{Apple, Banana})

Number of favourable outcomes =1= 1 Total number of outcomes =4= 4

P(Apple and Banana)=Number of favourable outcomesTotal number of outcomes=14\begin{aligned} P(\text{Apple and Banana}) &= \dfrac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}} \\[0.6em] &= \dfrac{1}{4} \end{aligned}
Answer

(iii) 14\dfrac{1}{4}

Common Mistakes
  • Incomplete Branches: Forgetting to branch each choice of Basket A into all possible choices of Basket B.
  • Outcome Miscount: Treating fruits from the two baskets as separate standalone outcomes rather than combined ordered pairs (A,B)(A, B).

More questions in Exercise 7.4

Q1

There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket.

(i) Draw a tree diagram showing all possible pairs of fruits.

(ii) List the sample space.

(iii) What is the probability of picking one apple and one banana?

Q2

Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same.

(i) What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes?

(ii) Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour?

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