To find the probability of rolling an even number exactly 5 times when rolling a six-sided number cube, we can use the binomial distribution.
### 1. Rolling a six-sided number cube 10 times
Given:
- Number of trials [tex]\( n = 10 \)[/tex]
- Probability of success on a single trial [tex]\( p = 0.5 \)[/tex] (since there are 3 even numbers out of 6 possible outcomes)
- Number of successes [tex]\( k = 5 \)[/tex]
Using the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Substitute the given values:
[tex]\[ P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^5 \][/tex]
After calculating, we get the probability:
[tex]\[ P(X = 5) \approx 0.246 \][/tex]
### 2. Rolling a six-sided number cube 20 times
Given:
- Number of trials [tex]\( n = 20 \)[/tex]
- Probability of success on a single trial [tex]\( p = 0.5 \)[/tex] (same reasoning as above)
- Number of successes [tex]\( k = 5 \)[/tex]
Using the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Substitute the given values:
[tex]\[ P(X = 5) = \binom{20}{5} (0.5)^5 (0.5)^{15} \][/tex]
After calculating, we get the probability:
[tex]\[ P(X = 5) \approx 0.015 \][/tex]
### Explanation of the result
The probability of rolling an even number exactly 5 times decreases when the number of trials increases from 10 to 20. This is because, with more trials, there are more possible distributions of outcomes, making the exact outcome of 5 successes among 20 trials (given the same probability of success) less likely compared to 5 successes among 10 trials.
In simpler terms, as you increase the number of trials, the outcomes become more spread out and achieving a specific exact count (like exactly 5 even numbers) becomes less probable.
