In an experiment, a fair coin is tossed 4 times and the number of heads that occur is recorded. Eighty trials are run.

The table shows the frequency of [tex]$0, 1, 2, 3,$[/tex] or [tex]$4$[/tex] heads occurring in the trials.
\begin{tabular}{|l|l|l|l|l|l|}
\hline Number of heads & 0 & 1 & 2 & 3 & 4 \\
\hline Frequency & 4 & 8 & 36 & 20 & 12 \\
\hline
\end{tabular}

Create a probability distribution for the discrete variable.

Probability Distribution of Heads:
\begin{tabular}{|l|l|l|l|l|l|}
\hline Number of heads & 0 & 1 & 2 & 3 & 4 \\
\hline Probability & [tex]$\frac{4}{80}$[/tex] & [tex]$\frac{8}{80}$[/tex] & [tex]$\frac{36}{80}$[/tex] & [tex]$\frac{20}{80}$[/tex] & [tex]$\frac{12}{80}$[/tex] \\
\hline
\end{tabular}



Answer :

Let's create a probability distribution for the discrete random variable representing the number of heads in four coin tosses, given the frequency data from the 80 trials.

### Step-by-Step Solution:

1. Understand the given frequency data:
- Number of heads: [tex]\( 0, 1, 2, 3 \)[/tex]
- Frequency: [tex]\( 4, 8, 36, 20 \)[/tex]

2. Calculate the total number of trials:
The total number of trials is the sum of the frequencies:
[tex]\[ \text{Total trials} = 4 + 8 + 36 + 20 = 68 \][/tex]

3. Determine the probability of each number of heads:
The probability for each number of heads is calculated by dividing the frequency of that outcome by the total number of trials.

- Probability of 0 heads:
[tex]\[ P(0) = \frac{\text{Frequency of 0 heads}}{\text{Total number of trials}} = \frac{4}{68} \approx 0.0588 \][/tex]

- Probability of 1 head:
[tex]\[ P(1) = \frac{\text{Frequency of 1 head}}{\text{Total number of trials}} = \frac{8}{68} \approx 0.1176 \][/tex]

- Probability of 2 heads:
[tex]\[ P(2) = \frac{\text{Frequency of 2 heads}}{\text{Total number of trials}} = \frac{36}{68} \approx 0.5294 \][/tex]

- Probability of 3 heads:
[tex]\[ P(3) = \frac{\text{Frequency of 3 heads}}{\text{Total number of trials}} = \frac{20}{68} \approx 0.2941 \][/tex]

4. Construct the probability distribution:

The probability distribution for the discrete variable, representing the number of heads, can now be summarized as follows:

| Number of heads ([tex]\(k\)[/tex]) | Probability ([tex]\(P(k)\)[/tex]) |
|------------------------|--------------------------|
| 0 | 0.0588 |
| 1 | 0.1176 |
| 2 | 0.5294 |
| 3 | 0.2941 |

The probabilities should sum to 1 (considering any rounding errors).

By setting the sliders on a probability distribution plot to the above-calculated probabilities, you will successfully represent the probability distribution of getting [tex]\(0, 1, 2, 3\)[/tex] heads in four coin tosses based on the given experimental data.