The Mathematics of Maybe: Introduction to Probability | Exercise 7.2

Question 1

A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour: 10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets

(i) Calculate the probability that a randomly picked sweet from the sample is green.

(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.

Solution

Understand the Question

The total number of sweets in the sample is $10 + 8 + 7 + 5 = 30$ .
The probability of an outcome is calculated using the formula: $P(E) = \dfrac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$
To estimate the expected number of a specific item in a larger population, multiply the sample probability by the total population size ( $N = 600$ ).

(i) Calculate the probability that a randomly picked sweet from the sample is green.

Step 1 · Calculate the Probability of Picking a Green Sweet

Total number of sweets in the sample $= 30$ Number of green sweets $= 8$ Diagram 1

\begin{aligned} P(\text{green sweet}) &= \dfrac{\text{Number of green sweets}}{\text{Total number of sweets}} \\[0.6em] &= \dfrac{8}{30} \\[0.6em] &= \dfrac{4}{15} \end{aligned}

Answer

(i) $\dfrac{4}{15}$

(ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results.

Step 1 · Estimate the Number of Yellow Sweets

First, find the probability of picking a yellow sweet from the sample:

\begin{aligned} P(\text{yellow sweet}) &= \dfrac{\text{Number of yellow sweets}}{\text{Total number of sweets in sample}} \\[0.6em] &= \dfrac{7}{30} \end{aligned}

Now, estimate the total number of yellow sweets in the bag of $600$ sweets:

\begin{aligned} \text{Estimated yellow sweets} &= P(\text{yellow sweet}) \times \text{Total sweets in bag} \\[0.6em] &= \dfrac{7}{30} \times 600 \\[0.6em] &= 7 \times 20 \\[0.6em] &= 140 \end{aligned}

Answer

(ii) $140$

Common Mistakes

Not Simplifying the Fraction: Leaving $\dfrac{8}{30}$ unsimplified instead of reducing it to lowest terms ( $\dfrac{4}{15}$ ).
Using the Wrong Denominator: Dividing by the total population size ( $600$ ) instead of the sample size ( $30$ ) when finding the sample probability.

Question 1

More questions in Exercise 7.2