An olympic archer is able to hit the bulls eye 80 of the...

An Olympic archer is able to hit the bull’s-eye 80% of the time. Assume each shot is independent of the others. If she shoots 6 arrows, what’s the probability of each of the following results? a) Her first bull’s-eye comes on the third arrow. b) She misses the bull’s-eye at least once. c) Her first bull’s-eye comes on the fourth or fifth arrow. d) She gets exactly 4 bull’s-eyes. e) She gets at least 4 bull’s-eyes. f) She gets at most 4 bull’s-eyes.( solve using R)

Transcript

00:01 An olympic archer hits the bolzai 80 % of the time.

00:05 Each shot is independent of the others.

00:08 So the first piece of information we have here is the probability of success on each trial is p equals 0 .8.

00:21 Okay, so we want some probabilities here and we're using r.

00:27 So when you're using r for your probability distributions, each distribution gets four functions.

00:35 They are p for probability, q for quantile, d for density, r for random.

00:41 Since we're using discrete probability distributions here, we're using the binomial and the geometric.

00:47 The ones you might be using are p binomial, so p binom.

00:58 And d binom, and then p, geom for geometric, and d, geom.

01:10 So the p is the cdf, so the cumulative frequency.

01:15 The d is the pdf, so the exact frequency for a single value of x.

01:23 Okay, so recap on these two functions before we dive in.

01:30 A geometric is defined as a series of failures.

01:34 Followed by a success.

01:36 And the probability of x being the first success means you have x minus one failures, so you multiply the failure by itself, x minus one times, and then you have one success.

01:52 Binomial, you have the additional parameter n, the number of trials, and probability of x successes in n trials is n choose x, p to be x, 1 minus p to be n minus x.

02:13 Now let's look at part a.

02:16 The first success is on the third arrow.

02:21 So this is going to be using the d -giom function in r, because i want the probability that x equals 3.

02:36 So i'm going to need to put in d -gium, and then 3, and the probability 0.

02:46 0 .8, like so.

02:49 3 being the value we're looking for.

02:52 Okay, so that is 0 .0 .032.

03:06 Now we want the probability of missing the bolzai at least once.

03:13 Okay, so we're going to be using the cumulative binomial here, and we want the probability that, but, okay, so she's shooting six arrows, so n equals six.

03:33 And we want the probability that she misses the bullseye at least once, so she has to have five or fewer successes.

03:45 So we're using the cumulative function here, and we want five, and then we also need the values n and p so n comes next, then p comes last.

04:03 Make some space for this.

04:13 Okay, so i need to throw that into r to get the answer...

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