What is the probability that her Internet service is not broken for four days in a row?

Question

Ashley’s Internet service is terribly unreliable. In fact, on any given day, there is an 80%80% chance that her Internet connection will be lost at some point that day. What is the probability that her Internet service is not broken for four days in a row?

Answer 1

Hi Amber,

1. Write out any values/information that we might be interested in.
2. 80% = the probability that Ashley's internet service is bad
3. In this case we can denote P(A) (the probability of event A occurring where "A" is defined as bad internet service) as 0.80.
4. So P(A) = 0.80
5. Write out a probability statement.
6. A probability statement is a written expression of what I am trying to find.
7. P ( Ashley's service is good for four days) --> Which is essentially saying, the probability that Ashley's service is good for four days
8. Determine the probability of interest.
9. In this case, we are interested in the probability that Ashley's service is not bad (or good).
10. Since we know that P(A) = 0.80 (the probability that Ashley's service is bad), we can use the complement rule to find the probability we are interested in.
11. The Complement Rule: States that the sum of the probability of event A occurring (where event A can be defined as anything) and the complement should be 1. Equation: P(A) + P(A') = 1.
12. Because we have one probability already (0.80), we can use the complement rule to find the probability of having good service. So we subtract 1 - 0.80 and get an answer of 0.20. (Equation used: 1 - P(A) = P(A')
13. 0.20 = the probability that Ashley has good service
14. P(A') = 0.20
15. Use multiplication to get a final answer.
16. Because we want to know the probability that Ashley has good service for four days, we can think as each day as an individual trial. In this case, we will use multiplication
17. P ( good service for 4 days) = 0.20 * 0.20 * 0.20 * 0.20 = 0.20⁴ = 0.0016

The probability that Ashley has good Internet service for 4 days in a row is 0.0016.