At peak periods 15 of attempted logins to an email service fail. log-in attempts are independent and each has the name probability of failing. Ann logs in repeatedly until she succeeds A. Find the probability that Ann needs to log in atleast four times before she succeeds
B. What is a standard deviation for number of logs-in needed to succeed.
To find the probability that Ann needs to log in at least four times before succeeding, we can use the concept of geometric distribution. The geometric distribution models the number of failures before the first success in a sequence of independent Bernoulli trials.
Let's denote the probability of a login attempt failing as p. Since the log-in attempts are independent, the probability of a login attempt succeeding is 1 - p.
A. Probability that Ann needs to log in at least four times before succeeding:
We need to calculate the probability of having 3 or more failures before the first success. This is equal to the sum of the individual probabilities of having 3, 4, 5, ... failures before the first success.
P(at least 4 attempts) = P(3 failures) + P(4 failures) + P(5 failures) + ...
Since each attempt follows a geometric distribution, the probability of having k failures before the first success is given by:
P(k failures) = p^k * (1 - p)
Therefore, the probability of having at least 4 attempts before succeeding is:
P(at least 4 attempts) = p^3 * (1 - p) + p^4 * (1 - p) + p^5 * (1 - p) + ...
We can express this as a sum of an infinite geometric series:
P(at least 4 attempts) = p^3 * (1 - p) * [1 + p + p^2 + p^3 + ...]
The sum of an infinite geometric series (1 + r + r^2 + r^3 + ...) with |r| < 1 is equal to 1 / (1 - r). In this case, r = p.
P(at least 4 attempts) = p^3 * (1 - p) * [1 / (1 - p)]
Simplifying this expression, we get:
P(at least 4 attempts) = p^3
Therefore, the probability that Ann needs to log in at least four times before succeeding is p^3.
B. Standard deviation for the number of log-ins needed to succeed:
The standard deviation of a geometric distribution is given by the formula:
Standard deviation = sqrt((1 - p) / (p^2))
Therefore, the standard deviation for the number of log-ins needed to succeed is sqrt((1 - p) / (p^2)).