The Mathematics of Maybe: Introduction to Probability | Exercise 7.2

Question 3

Toss a coin 20 times and record the result each time (heads or tails).

(i) How many times did you get heads?

(ii) How many times did you get tails?

(iii) Calculate the experimental probability of getting heads.

(iv) If you toss the coin once more, what is the probability of getting tails?

Solution

Understand the Question

This is an activity-based question based on experimental probability and independent events.
Experimental Probability is calculated using the observed outcomes: $\text{Experimental Probability} = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of trials}}$
For any single toss of a fair coin, the outcomes (heads or tails) are equally likely, so the theoretical probability of getting tails is always $\dfrac{1}{2} = 0.5$ .
(Note: In this sample solution, we consider an experiment where tossing a coin $20$ times resulted in $11$ heads and $9$ tails. Actual values may vary based on your experiment.)

(i) How many times did you get heads?

Step 1 · Record Number of Heads

From the experiment of $20$ tosses, suppose heads appears $11$ times. Diagram 1

$\text{Number of heads} = 11$

Answer

(i) $11$

(ii) How many times did you get tails?

Step 1 · Calculate Number of Tails

\begin{aligned} \text{Number of tails} &= \text{Total tosses} - \text{Number of heads} \\[0.6em] &= 20 - 11 \\[0.6em] &= 9 \end{aligned}

Answer

(ii) $9$

(iii) Calculate the experimental probability of getting heads.

Step 1 · Calculate Experimental Probability

\begin{aligned} P(\text{Heads}) &= \dfrac{\text{Number of heads}}{\text{Total number of tosses}} \\[0.6em] &= \dfrac{11}{20} \\[0.6em] &= 0.55 \end{aligned}

Answer

(iii) $0.55$

(iv) If you toss the coin once more, what is the probability of getting tails?

Step 1 · Find Probability of Tails for an Independent Toss

Each toss of a fair coin is independent of previous results. The two outcomes ( $\{\text{Heads}, \text{Tails}\}$ ) are equally likely.

$P(\text{Tails}) = \dfrac{1}{2} = 0.5$

Answer

(iv) $0.5$

Common Mistakes

Gambler's Fallacy: Believing that previous results influence the next toss. Each toss is independent, so the theoretical probability of getting tails on a fresh toss remains $\dfrac{1}{2} = 0.5$ .
Confusing Experimental and Theoretical Probability: Experimental probability depends directly on the observed data ( $\dfrac{11}{20}$ here), whereas the single-toss theoretical probability is $\dfrac{1}{2}$ regardless of past results.

More questions in Exercise 7.2