Probability of Hitting a Target The probability that an...

Probability of Hitting a Target The probability that an archer hits the target is $p=0.9,$ so the probability that he misses the target is $q=0.1 .$ It is known that in this situation the probability that the archer hits the target exactly $r$ times in $n$ attempts is given by the term containing $p^{\prime}$ in the binomial expansion of $(p+q)^{n}$ . Find the probability that the archer hits the target exactly three times in five attempts.

Probability of Hitting a Target The probability that an archer
hits the target is $p=0.9,$ so the probability that he misses
the target is $q=0.1 .$ It is known that in this situation the
probability that the archer hits the target exactly $r$ times in $n$ attempts is given by the term containing $p^{\prime}$ in the binomial
expansion of $(p+q)^{n}$ . Find the probability that the archer
hits the target exactly three times in five attempts.

Key Concepts

Binomial Distribution

A probability distribution that describes the likelihood of obtaining a fixed number of successes in a specified number of independent Bernoulli trials, each with the same probability of success. It is widely used for modeling situations where there are exactly two outcomes per trial.

Bernoulli Trial

A random experiment with exactly two possible outcomes: success or failure. Each trial is independent, and the probability of success remains constant, making it the basic component of the binomial distribution.

Binomial Coefficient

A combinatorial term that calculates the number of ways to choose a specific number of successes from a total number of independent trials. It is denoted as C(n, r) or nCr and is critical in determining the weighting of each term in the binomial probability formula.

Binomial Theorem

A mathematical principle that provides a systematic method for expanding powers of binomials. In probability, it underpins the derivation of the binomial distribution by showing how the probabilities of all possible outcome combinations add up to one.

Transcript

00:01 All right, we got a problem with context, and i always really do enjoy problems with context.

00:06 It makes me feel like there is definitely applicable purpose to everything that we do.

00:10 So i'm assuming that you've read all this.

00:12 The important parts is that the probability that archer hits the target is 0 .9 or 90%, leaving a 10 % or 0 .1 as a decimal probability of missing the target.

00:23 And here's the thing that i think is a little bit weird.

00:27 I think i can understand, okay, they're going to hit it r times and n attempts.

00:31 I can understand that.

00:32 I struggled with what does it mean that the probability of hitting it is p raised the r in the binomial expansion? well, let's go ahead and do that binomial expansion.

00:46 P plus q raised to the, well, it's going to be raised to the number of temps, the fifth power.

00:57 And i do know what p is and i do know what q is.

00:59 But i'm going to go ahead and just leave it as p and q for the expansion because it's easy to do.

01:04 This is p, raises a fifth.

01:05 Of course, one is the coefficient.

01:08 And i'm going to do just the slightest bit of cheating.

01:11 I know using pascal's triangle that the coefficient for this next one is five, as opposed to using the binomial theorem.

01:19 I mean, i feel like this problem, i'm going to just go ahead and get away with that because that's not the point of finding the coefficient.

01:26 It's happened out of the problems.

01:27 Okay.

01:28 P is now to the fourth power.

01:30 To the first power using pascal's triangle.

01:33 The next coefficient is 10.

01:36 P's to the third, q is to the second, and another 10 for the coefficient.

01:43 P is now to the second and q's to the third.

01:47 Some of these are actually going to be kind of irrelevant, and these are all relevant ones, but i'll say why.

01:52 5.

01:53 P is to the first power, q is to the fourth, and one is the coefficient for the last term where q is to the fifth.

02:02 And let me explain what this expansion shows...

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