A random number generator draws at random with replacement from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Find the chance that the digit 5 appears on more than 11% of the draws, if
a) 100 draws are made
b) 1000 draws are made
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a)0.2969
b)0.1347765
To find the chance that the digit 5 appears on more than 11% of the draws, we need to calculate the probability of getting more than a certain number of 5's in a given number of draws.
Let's start with part a) where 100 draws are made.
a) 100 draws are made:
To calculate the probability of getting more than 11% 5's in the 100 draws, we need to determine the number of draws for which there are more than 11 5's.
The probability of getting a 5 in any single draw is 1/10 (since there are 10 digits to choose from).
Now, let's calculate the probability of getting exactly 11 5's in the 100 draws. This can be calculated using the binomial probability formula:
P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k))
where:
- P(X = k) is the probability of getting exactly k successes (in this case, 11 5's)
- nCk is the binomial coefficient representing the number of ways to choose k successes out of n total trials
- p is the probability of success in a single trial (1/10)
- k is the number of successes (11 in this case)
- n is the total number of trials (100 draws)
Plugging in the values into the formula:
P(X = 11) = (100C11) * ((1/10)^11) * ((9/10)^(100-11))
Next, we need to calculate the probability of getting 12, 13, 14, ..., up to 100 5's. We can then sum up these probabilities to get the final probability of getting more than 11% 5's.
Finally, we can subtract the sum of these probabilities from 1 to get the probability of getting more than 11% 5's:
P(more than 11% 5's) = 1 - (P(X = 11) + P(X = 12) + P(X = 13) + ... + P(X = 100))
b) 1000 draws are made:
The calculation for part b) is similar to part a). We would need to calculate the probabilities for getting 11, 12, 13, ..., up to 1000 5's, and then sum up these probabilities to find the chance of getting more than 11% 5's.
We first calculate the probability of getting exactly X number of 5's using the binomial probability formula, and then sum up the probabilities for X = 11 to X = 1000. Finally, we subtract the sum from 1 to get the probability of getting more than 11% 5's.
Note: The calculations involved in finding the exact probabilities for each X can be quite complex, especially when dealing with a large number of draws like in part b) where 1000 draws are made. In practice, numerical methods or simulation methods might be used to estimate these probabilities.