Archery An archer hits his target 80% of the time. If he...

Archery An archer hits his target 80$\%$ of the time. If he shoots 7 arrows, what is the probability of each event? (a) He never hits the target. (b) He hits the target each time. (c) He hits the target more than once. (d) He hits the target at least 5 times.

Transcript

00:01 In this question, we're given that archer hits his target 80 % of the time and that he shoots seven arrows.

00:08 I'm going to let x be the number of hits out of seven shoots.

00:12 X follows the binomial distribution.

00:14 Let's see how it does that.

00:17 The binomial distribution has four criteria.

00:20 The first criteria, the number of trial, n, is fixed and are identical.

00:25 In this case, we have seven shoots.

00:28 And in each of this shoot, we are looking at whether he hits the target or not.

00:36 So the trial as in the experiment is looking at whether every shoot, whether he hits the target.

00:43 So they are identical and it speaks at seven.

00:47 So first criteria is fulfilled.

00:49 Second criteria, each trial is independent.

00:52 Now for each shoot, whether he hit the target, not it's independent each other.

00:56 So the second criteria is fulfilled.

00:59 Third criteria, each trial results in one of two outcomes, success and failure.

01:05 So for each shoot, a success would be he hit the target, failure will be he did not hit the target.

01:12 So the third criteria is fulfilled.

01:14 Now, the last criteria, probability of success p, remains the same from trial to trial.

01:20 Now in this case, for each shoot, the probability of him hitting the target is 80%, or 0 .8 in decimal.

01:28 So the fourth criteria is fulfilled.

01:31 So my p is 0 .8.

01:36 Now you can see that my x follows the binomial distribution.

01:41 My n is 7 and p is 0 .8.

01:44 So probability of x equals to r, where r is the number of heats out of seven shoots, will be 7 choose r, 0 .8 to power r, and 1 minus 0 .8 will be 0 .2 to the power of 7 minus r.

01:58 Now in part a, we want to find probability he never hits the target.

02:04 So that would be probability x equals to 0.

02:09 So just sub -zero into the r.

02:19 And you will get 0 .00128.

02:27 And that is 0 .0004 decimal place...

Question

Archery An archer hits his target 80$\%$ of the time. If he shoots 7 arrows, what is the probability of each event? (a) He never hits the target. (b) He hits the target each time. (c) He hits the target more than once. (d) He hits the target at least 5 times.

Please give Ace some feedback