Question

if your sms text message fails to send 1 message for every 10,000 messages you send, how likely is it that only 1 message of yours failed from a sample of 1,000 messages?

Answer 1

The probability of exactly one message failing out of 1,000 messages, given that on average one fails out of every 10,000, is approximately 9.05%.

Step-by-step explanation:

The scenario given is a probability question where we need to calculate the likelihood of a specific event happening under given conditions.

According to the information provided, if 1 message fails for every 10,000 messages sent, we can deduce that the probability of one message failing is 0.0001 (which is the probability of failure, p).

Let's calculate the probability of exactly 1 message failing out of 1,000 sent messages using the binomial probability formula P(X=k) = C(n, k)*(p^k)*((1-p)^(n-k)), where C(n, k) is the combination of n items taken k at a time:

Since the probability of a text message failing is quite small, and we are looking at a relatively small number of trials (1,000 messages), this situation can be approximated using the Poisson distribution.

The average number of failures we can expect in 1,000 messages (lambda) is (1,000 messages)*(0.0001 failure rate) = 0.1 failure.

Using the Poisson probability formula P(X=k) = (e^(-lambda) * lambda^k) / k!:

1. Calculate lambda: lambda = 1,000 * 0.0001 = 0.1
2. Compute the probability for k=1 failure: P(X=1) = (e^(-0.1) * 0.1^1) / 1! = 0.0905 (approximately)

The probability that only 1 message out of 1,000 messages fails is approximately 9.05%.