Question

Imagine tossing a fair coin $3$ times. What is the sample space for this chance process and what is the assignment of probabilities to outcome in this sample space?

Answer 1

Step 1 : Explain the sample space:

The set of all possible outcomes of an experiment is called the sample space of the experiment. The symbol, ‘S denotes it.’

Sample space for tossing $3$fair coins:

The possible outcomes of tossing a coin are head and tail.

Let $\mathrm{H}$ denotes head and $\mathrm{T}$ denote tail.

The sample space for tossing $3$fair coins is:

$\mathrm{S}=\{\mathrm{HHH},\mathrm{TTT},\mathrm{HHT},\mathrm{HTH},\mathrm{THH},\mathrm{TTH},\mathrm{THT},\mathrm{HTT}\}$

Step 2 : Find the probability:

The number of outcomes in the sample space is $8$.

There are $8$ possible events.

Since the coin is fair, each of the $8$ events is possible.

All the events in the sample space have the same probability.

Probability, $\mathrm{P}=\frac{1}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{outcomes}\mathrm{in}\mathrm{sample}\mathrm{space}}$

$\Rightarrow \mathrm{P}=\frac{1}{8}$

Hence, the sample space is, $\mathrm{S}=\{\mathrm{HHH},\mathrm{TTT},\mathrm{HHT},\mathrm{HTH},\mathrm{THH},\mathrm{TTH},\mathrm{THT},\mathrm{HTT}\}$ and the probability of outcomes in the sample space is $\frac{1}{8}$.