At peak periods, 15% of attempted log-ins to an email service fail. Log-ins attempts are independent and each has the same probability of failing. Ann logs in repeatedly until she succeeds.
A. find the probability that Ann needs to log in at least four times before she succeeds.
B. What is a standard deviation for number of log-ins needed to succeed
Please HELP!!!Thank you so much!
To solve this problem, we can use the concept of geometric probability, since Ann keeps trying until she succeeds.
A. Probability that Ann needs to log in at least four times before she succeeds:
Let's denote the probability of success (i.e., a successful log-in) as p = 0.85 (since 15% of log-ins fail, the successful log-in probability is 1 - 0.15 = 0.85).
The probability that Ann succeeds on the first log-in attempt is p = 0.85.
The probability that Ann succeeds on the second log-in attempt is (1 - p) * p = 0.15 * 0.85 = 0.1275.
The probability that Ann succeeds on the third log-in attempt is (1 - p) * (1 - p) * p = 0.15 * 0.15 * 0.85 = 0.018225.
The probability that Ann succeeds on the fourth or later log-in attempt is (1 - p) * (1 - p) * (1 - p) = 0.15 * 0.15 * 0.15 = 0.003375.
To find the probability that Ann needs to log in at least four times before she succeeds, we sum the probabilities of these scenarios:
P(Ann needs to log in at least four times before she succeeds) = 0.018225 + 0.003375 = 0.0216
Therefore, the probability that Ann needs to log in at least four times before she succeeds is 0.0216 or 2.16%.
B. Standard deviation for the number of log-ins needed to succeed:
To find the standard deviation, we use the formula for the geometric distribution: σ = √(q/p^2), where q = 1 - p.
σ = √(0.15 / 0.85^2) ≈ 0.396
Therefore, the standard deviation for the number of log-ins needed to succeed is approximately 0.396.
To answer the given questions, we need to understand the concept of a geometric distribution, which is the probability distribution of the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials (where each trial has only two possible outcomes: success or failure).
The geometric distribution is used when each trial has the same probability of success, and the trials are independent of each other. In this case, the log-in attempts have the same probability of failing, and they are independent.
Let's solve the questions using this information:
A. Find the probability that Ann needs to log in at least four times before she succeeds:
To find the probability that Ann needs to log in at least four times before she succeeds, we need to calculate the sum of probabilities of failing for the first three attempts and succeeding on the fourth attempt.
The probability of failing on each attempt is 15%, which means the probability of success on each attempt is 100% - 15% = 85%.
The probability that Ann succeeds on the fourth attempt is (0.85)^3 * 0.15, as she needs three failures (0.85^3) and then a success (0.15).
So, the probability that Ann succeeds on at least the fourth attempt is 0.85^3 * 0.15 = 0.09621.
B. Find the standard deviation for the number of log-ins needed to succeed:
To calculate the standard deviation for a geometric distribution, we need to use the following formula:
Standard Deviation = sqrt((1 - p) / p^2),
where p is the probability of success (0.15 in this case).
Applying the formula, we get:
Standard Deviation = sqrt((1 - 0.15) / 0.15^2) = sqrt(0.85 / 0.0225) ≈ sqrt(37.78) ≈ 6.14.
Therefore, the standard deviation for the number of log-ins needed to succeed is approximately 6.14.
I hope this explanation helps! If you have any further questions, feel free to ask.