7. A Continuously running washing machine receives jolts in a Poisson process at an average rate of 5 jolts per hour. The probability that it receives no jolts in given 20 minute run
Choose one answer
a. e−5
b. e−5/3
c. e−100
d. e−100/3
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e−100/3
To solve this problem, we can use the Poisson distribution. The Poisson distribution models the number of events that occur in a fixed interval of time or space when these events happen with a known average rate and independently of the time since the last event.
The average rate of jolts is given as 5 jolts per hour. We want to find the probability that the washing machine receives no jolts in a 20-minute (1/3 hour) run.
Let's denote λ as the average rate of jolts per hour. In this case, λ = 5.
The probability of receiving no jolts in a given time period can be calculated using the Poisson distribution formula:
P(x = 0) = (e^(-λ) * λ^x) / x!
Where:
- x is the number of events (in this case, the number of jolts)
- e is the base of the natural logarithm, approximately 2.71828
In our case, x = 0 (we want to find the probability of receiving no jolts), and λ = 5/3 (since the 20-minute run is 1/3 of an hour).
Substituting these values into the formula, we get:
P(x = 0) = (e^(-5/3) * (5/3)^0) / 0!
Since any number raised to the power of 0 is equal to 1, and 0! (0 factorial) is equal to 1, the formula simplifies to:
P(x = 0) = e^(-5/3)
So, the answer to the question is b. e^(-5/3).