2 the reliability of power stations on the electricity grid...

2. The reliability of power stations on the electricity grid is quantified by their Availability Factor, which is the proportion of time on average for which the station is online (i.e. operating and able to produce power). In Ireland, the average Availability Factor for power stations is estimated at 0.85, or 85%. Assume that all power stations on the grid are identical, and that the availability of each station is completely independent of the other stations. If a sample of 8 Irish power stations is chosen at random: (a) What is the probability that exactly one station will be offline (i.e. unavailable)? [5] (b) What is the probability that at least 6 stations in the sample will be online (i.e. available)? [10] (c) What is the probability that between 5 and 7 stations will be online? [5] (d) What is the sample size for which is expected that 3 stations in the sample will be offline? [5] [Total 25

Transcript

00:01 In this question, it is given that the number of power outages at nuclear power plant has position with a mean of, mean is given as six outages per year.

00:12 So we need to tell what is the probability that there will be exactly three power outages in a year.

00:20 So number of power outages are exactly three.

00:24 So we need to answer the probability for that condition.

00:29 So we can assume let x denotes the number of power outages, then there are, if there are exactly three power outages, so we need to find the probability when there is exactly three power outages.

00:58 And that can be written as e -raise to the power minus lambda times lambda -raise to the power x divided by x factorial so from here we can say that the answer will be lambda in place of lambda we will put the mean and mean is given as six so we can say that probability of exactly three power outages will be a raise to the power minus lambda is six and it is six raised to the power their x will be three divided by three factorial so from here if we calculate this value we will be getting 0.

01:42 So, 892.

01:46 The probability of getting exactly three power outages in a year is 0 .089.

01:55 This will be the answer to the first part.

01:59 This is a part.

02:01 Now in the next part it is being asked, in the next part it is being asked that we need to find out the probability that there will be at least three power outages in a year.

02:12 So for at least three power outages, what we will do? we need to find out the probability that x is greater than equals to three because we are going to answer for at least three power outages and that will be one minus probability of when there is no power outage minus probability when there is one power outage minus probability when there is two power outages so we need to find out this answer so what we can write is this will be equals to 1 minus e raise to the power minus 6 times 6 raise to the power 0 divided by 0 factorial minus e raise to the power minus 6 times 6 raised to the power 1 divided by 1 factorial minus e to the power minus 6 times 6 raise to the power 2 divided by 2 factorial...

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