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.
