Find the probability of getting a prime number thrice by tossing a die 10 times.
Well, rolling a die to get a prime number isn't exactly prime material for comedy, but let's give it a shot!
To find the probability of getting a prime number thrice by tossing a die 10 times, we first need to know how many prime numbers appear when rolling a die. So, let's count them: 2, 3, and 5. Yup, that's right, no other prime numbers on a regular die!
Now, let's calculate the probability. Since there are three prime numbers on the die and a total of six possible outcomes, the probability of rolling a prime number is 3/6, which simplifies to 1/2.
Since we want to roll a prime number thrice, we multiply 1/2 three times, like this: (1/2) x (1/2) x (1/2).
Calculating that brings us to the final answer: 1/8.
So, the probability of getting a prime number thrice by tossing a die 10 times is 1/8. Just make sure to toss with confidence and a sense of mathematical whimsy!
To find the probability of getting a prime number thrice by tossing a die 10 times, we need to determine two things:
1. The total number of possible outcomes: Since we are tossing a die 10 times, each toss has 6 possible outcomes (numbers 1 to 6). Therefore, the total number of possible outcomes for 10 tosses is 6^10 (6 multiplied by itself 10 times).
2. The number of favorable outcomes: We need to count the number of outcomes where a prime number appears exactly three times in the 10 tosses.
To do this, we can approach it using the concept of combinations.
Step 1: Determine the possible locations of the prime numbers.
- There are a total of 10 tosses, and we need 3 of them to be a prime number.
- This can be represented as selecting 3 locations out of 10. The number of ways to do this is given by the combination formula: C(10,3) = 10! / (3! * (10-3)!)
- Evaluating this gives C(10,3) = 120.
Step 2: For each of these 3 locations, we have 2 options (the prime numbers are 2, 3, 5).
- So, the three favorable locations can be filled in 2^3 ways.
Step 3: The remaining locations (7 tosses) need to be filled with any of the remaining 3 non-prime numbers (1, 4, and 6), which gives us 3^7 ways.
Step 4: Now, to find the total number of favorable outcomes, we multiply the values from steps 2 and 3.
- The number of favorable outcomes = 2^3 * 3^7 = 1728.
Therefore, the probability of getting a prime number thrice by tossing a die 10 times is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
= 1728 / (6^10)
≈ 0.0000151
So, the probability is approximately 0.0000151 or 1.51 × 10^(-5).
miho miho
On a die, 3 numbers are prime (2, 3, 5) and 3 numbers are not prime (1, 4, 6).
Since the chances of rolling a prime each time are the same as those of not rolling a prime each time, we can treat it much like a coin toss.
2 outcomes each roll (Prime or Not Prime), with 10 rolls, gives a total number of possible outcomes of 2^10 = 1024.
In 10 rolls, 3 Primes can occur in 10c3 ways.
Therefore probability is 10c3/(2^10).