Question

A computer is used to generate passwords made up of numbers 0 through 9 and lowercase letters. The computer generates 400 passwords one character at a time.
A uniform probability model is used to predict the first character in the password.

What is the prediction for the number of passwords in which the first character is a vowel?

Round your answer to the nearest whole number.


56 passwords

77 passwords

111 passwords

233 passwords

Answer 1

The prediction for the number of passwords in which the first character is a vowel is 56 passwords.

How to find that a given condition can be modelled by binomial distribution?

Binomial distributions consist of n independent Bernoulli trials.

Bernoulli trials are those trials which end up randomly either on success (with probability p) or on failures( with probability 1- p = q (say))

Suppose we have random variable X pertaining to a binomial distribution with parameters n and p, then it is written as

$X \sim B(n,p)$

The probability that out of n trials, there'd be x successes is given by

$P(X =x) = \: ^nC_xp^x(1-p)^{n-x}$

The expected value and variance of X are:

$E(X) = np$

Given that the characters that can be used are numbers 0 through 9 and lowercase letters. Therefore, a total of 36 different characters are available.

Since we need to know the passwords made with vowels, therefore, the probability of a password in which the first character will be a, e, i, o, u is (5/36).

Now as the computer produces 400 passwords, therefore, the predicted value can be written as,

$E = np = 400 \times \dfrac{5}{36} = 55.5556 \approx 56$

Hence, the prediction for the number of passwords in which the first character is a vowel is 56 passwords.

Learn more about Binomial Distribution:

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