A random number generator draws at random with replacement from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In 5000 draws, the chance that the digit 0 appears fewer than 495 times is closest to 25%,30%,35%,40%,45%,50%
I know that 35% is wrong!!!
Take to Binomial Distn to Normal Distn. Si it becomes P(x<=494).
I know that 30% is also wrong!
It is 40% the correct answer
40% is correct.
To calculate the probability that the digit 0 appears fewer than 495 times in 5000 draws, we can use the concept of the binomial distribution.
The binomial distribution is a probability distribution that describes the number of successes (in this case, the appearance of the digit 0) in a fixed number of independent Bernoulli trials (in this case, the draws from the random number generator).
In this scenario, the probability of getting a digit 0 in any given draw is 1/10, since there are 10 possible digits. Therefore, the probability of not getting a digit 0 in any given draw is 1 - 1/10 = 9/10.
Now, we want to find the probability that the digit 0 appears fewer than 495 times in 5000 draws. We can apply the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of its complement (not occurring).
The complement of getting fewer than 495 occurrences of the digit 0 is getting 495 or more occurrences of the digit 0. So, we need to find the probability of getting 495, 496, 497, ..., 5000 occurrences of the digit 0.
Using the binomial distribution formula, the probability of getting exactly k occurrences of the digit 0 in 5000 draws is given by:
P(X = k) = C(5000, k) * (1/10)^k * (9/10)^(5000 - k)
where C(n, k) denotes the number of combinations of n items taken k at a time.
To find the probability of getting 495 or more occurrences of the digit 0, we can sum up the probabilities of getting 495, 496, 497, ..., 5000 occurrences:
P(X >= 495) = P(X = 495) + P(X = 496) + ... + P(X = 5000)
Calculating this sum can be computationally intensive, but we can use mathematical software or programming languages to do the calculations. In this case, it is not practical to perform the calculations manually.
Using a mathematical software like R, we can use the function `pbinom()` to calculate the cumulative probability. Here is an example of how we can do this in R:
```R
p <- sum(pbinom(495:5000, size = 5000, prob = 1/10, lower.tail = FALSE))
```
The variable `p` will contain the probability of getting 495 or more occurrences of the digit 0 in 5000 draws. Subtracting this probability from 1 will give us the probability of getting fewer than 495 occurrences:
```R
result <- 1 - p
```
Finally, to identify the closest option from the given choices, we can compare the calculated probability with the given percentages and select the closest one.