The monthly worldwide average number of airplane crashes of...

The monthly worldwide average number of airplane crashes of commercial airlines is $3.5 .$ What is the probability that there will be
(a) at least 2 such accidents in the next month;
(b) at most 1 accident in the next month? Explain vour reasoning!

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution used to model the number of events that occur within a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is widely used to represent scenarios such as accidents, arrivals, or occurrences that are rare in nature.

Parameter ?

In the context of the Poisson distribution, ? (lambda) represents the average rate or expected number of occurrences per interval. This parameter is crucial because it defines the distribution; for example, in scenarios where events happen on average 3.5 times in a month, ? would be 3.5.

Probability Mass Function (PMF)

The probability mass function (PMF) for the Poisson distribution gives the probability of observing exactly k events in a fixed interval. It is defined by the formula P(X = k) = (e^(-?) * ?^k) / k!, which is used to compute the likelihood of any specific count of events, such as zero, one, or two occurrences.

Complement Rule

The complement rule in probability is a method used to simplify calculations, especially when dealing with 'at least' scenarios. Instead of summing probabilities over many cases, one can calculate the probability of the complementary event (for example, fewer than a certain number of events) and subtract that value from one to obtain the desired probability.

Transcript

00:01 Hi guys in this problem we have probability of x is equal to 25 .2 % which means that it's 0 .252 and probability of y is equal to 23 .6%.

00:18 So this is 0 .236.

00:22 Okay so x follows a polynomial distribution with n is equal to 200 and p is equal 0 .252 and y is equal follows a polynomial distribution with n is equal to 200 and p is equal to 0 .236 okay so the mean of x is equal to n p which is 200 times 0 .252 so this is 50 .4 and the variance of x is equal to n p times 1 minus p so this is 50 .4 times 1 .252 so the variance of x is equal to 47 .2 okay so x here follows in normal distribution with a mean is equal to 50 .4 and standard deviation is equal to um is equal to what is equal to i'm sorry here it's um 37 .69 okay so here is 37 .69 then uh we need to find the mean of y so it's also mp which is 200 times 0 .236 which is 47 .3 6 which is 47 .2 and the variance of y, it's np 200 times 0 .236 times 1 minus 0 .236.

02:31 So this is 36 .06.

02:37 Okay.

02:38 So y also falls in normal distribution with mean is equal to 47 .2.

02:46 And a standard division is equal to, or the variance is equal to 36 .06.

02:55 Okay, then in part of e, we need to determine the probability that at least 110 of those 400 people never eat breakfast.

03:05 So here we have to find and calculate the probability of x and y more than would equal to 1009.

03:17 So we have the mean of x plus y.

03:26 It's just the mean of x plus the mean of y.

03:31 So this is equal to 97 .6...