Alright, let's look at the problem step by step.
Given:
A die is thrown 300 times, and the frequency distribution of the outcomes is as follows:
[tex]\[
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\text{Outcomes} & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\text{Frequency} & 30 & 35 & 55 & 50 & 60 & 70 \\
\hline
\end{tabular}
\][/tex]
We are to find the following probabilities:
(i) a number divisible by 3,
(ii) an odd number,
(iii) a prime number,
(iv) an even number,
(v) number divisible by 2.
Total number of trials (throws) = 300
Let's calculate each probability step by step.
### (i) Probability of a number divisible by 3
Numbers divisible by 3 from the die outcomes are 3 and 6.
- Frequency of 3 = 55
- Frequency of 6 = 70
Total frequency for numbers divisible by 3 = 55 + 70 = 125
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Total frequency of numbers divisible by 3}}{\text{Total number of trials}} = \frac{125}{300} = 0.4166666666666667 \][/tex]
### (ii) Probability of an odd number
Odd numbers from the die outcomes are 1, 3, and 5.
- Frequency of 1 = 30
- Frequency of 3 = 55
- Frequency of 5 = 60
Total frequency for odd numbers = 30 + 55 + 60 = 145
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Total frequency of odd numbers}}{\text{Total number of trials}} = \frac{145}{300} = 0.48333333333333334 \][/tex]
### (iii) Probability of a prime number
Prime numbers from the die outcomes are 2, 3, and 5.
- Frequency of 2 = 35
- Frequency of 3 = 55
- Frequency of 5 = 60
Total frequency for prime numbers = 35 + 55 + 60 = 150
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Total frequency of prime numbers}}{\text{Total number of trials}} = \frac{150}{300} = 0.5 \][/tex]
### (iv) Probability of an even number
Even numbers from the die outcomes are 2, 4, and 6.
- Frequency of 2 = 35
- Frequency of 4 = 50
- Frequency of 6 = 70
Total frequency for even numbers = 35 + 50 + 70 = 155
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Total frequency of even numbers}}{\text{Total number of trials}} = \frac{155}{300} = 0.5166666666666667 \][/tex]
### (v) Probability of a number divisible by 2
Numbers divisible by 2 from the die outcomes are 2, 4, and 6. Essentially, this is the same set of numbers as the even numbers.
Total frequency for numbers divisible by 2 = 35 + 50 + 70 = 155
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Total frequency of numbers divisible by 2}}{\text{Total number of trials}} = \frac{155}{300} = 0.5166666666666667 \][/tex]
In summary, the probabilities are:
1. Probability of a number divisible by 3: [tex]\(0.4166666666666667\)[/tex]
2. Probability of an odd number: [tex]\(0.48333333333333334\)[/tex]
3. Probability of a prime number: [tex]\(0.5\)[/tex]
4. Probability of an even number: [tex]\(0.5166666666666667\)[/tex]
5. Probability of a number divisible by 2: [tex]\(0.5166666666666667\)[/tex]
