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4. A die is thrown 300 times. The frequency distribution table having the frequencies of the outcomes of [tex]$1, 2, 3, 4, 5, 6$[/tex] is given below.

\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
Outcomes & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
Frequency & 30 & 35 & 55 & 50 & 60 & 70 \\
\hline
\end{tabular}

Find the probability of the occurrence of:

(i) A number divisible by 3

(ii) An odd number

(iii) A prime number

(iv) An even number

(v) A number divisible by 2


Sagot :

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]