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5. Prime numbers between 1 and 25 are written on identical slips, put in a box, and mixed well. If a slip is drawn at random, find the probability of getting:

i. a one-digit number
ii. an even number
iii. an odd number
iv. a number greater than 11

Sagot :

GinnyAnswer
Sure, let's find the probabilities step by step.

### Prime Numbers between 1 and 25

First, let's list out all the prime numbers between 1 and 25. They are:
[tex]\[ 2, 3, 5, 7, 11, 13, 17, 19, 23 \][/tex]

There are 9 prime numbers in total.

### i. Probability of Getting a One-Digit Number

The one-digit prime numbers are:
[tex]\[ 2, 3, 5, 7 \][/tex]

There are 4 one-digit prime numbers. Therefore, the probability [tex]\(P\)[/tex] of drawing a one-digit prime number is:
[tex]\[ P(\text{one-digit number}) = \frac{\text{Number of one-digit primes}}{\text{Total number of primes}} = \frac{4}{9} \approx 0.4444444444444444 \][/tex]

### ii. Probability of Getting an Odd Number

The odd prime numbers are:
[tex]\[ 3, 5, 7, 11, 13, 17, 19, 23 \][/tex]

There are 8 odd prime numbers. Therefore, the probability [tex]\(P\)[/tex] of drawing an odd prime number is:
[tex]\[ P(\text{odd number}) = \frac{\text{Number of odd primes}}{\text{Total number of primes}} = \frac{8}{9} \approx 0.8888888888888888 \][/tex]

### iii. Probability of Getting an Even Number

The even prime number is:
[tex]\[ 2 \][/tex]

There is 1 even prime number. Therefore, the probability [tex]\(P\)[/tex] of drawing an even prime number is:
[tex]\[ P(\text{even number}) = \frac{\text{Number of even primes}}{\text{Total number of primes}} = \frac{1}{9} \approx 0.1111111111111111 \][/tex]

### iv. Probability of Getting a Number Greater than 11

The prime numbers greater than 11 are:
[tex]\[ 13, 17, 19, 23 \][/tex]

There are 4 prime numbers greater than 11. Therefore, the probability [tex]\(P\)[/tex] of drawing a prime number greater than 11 is:
[tex]\[ P(\text{greater than 11}) = \frac{\text{Number of primes greater than 11}}{\text{Total number of primes}} = \frac{4}{9} \approx 0.4444444444444444 \][/tex]

### Summary of Results

- Probability of getting a one-digit number: [tex]\( \approx 0.4444444444444444 \)[/tex]
- Probability of getting an odd number: [tex]\( \approx 0.8888888888888888 \)[/tex]
- Probability of getting an even number: [tex]\( \approx 0.1111111111111111 \)[/tex]
- Probability of getting a number greater than 11: [tex]\( \approx 0.4444444444444444 \)[/tex]

These probabilities are based on the total of 9 prime numbers between 1 and 25.
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