Archery An archer hits his target 80% of the time. If he shoots seven arrows, what is the probab

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

Key Concepts

Cumulative Probability

Cumulative probability is used when the problem asks for the probability of achieving a range of outcomes, such as hitting the target 'more than once' or 'at least five times.' It involves summing individual binomial probabilities across the specified range of successes.

Bernoulli Trials

This concept refers to experiments with exactly two outcomes, often termed as 'success' and 'failure', where each trial is independent and has the same probability of success. In the context of the problem, each arrow shot is a Bernoulli trial since it results in either a hit or a miss.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials. It is used when calculating the probability of achieving a specific number of hits (successes) out of a set number of shots, using the formula that incorporates binomial coefficients and the probabilities of success and failure.

Combination (Binomial Coefficient)

This concept involves calculating the number of ways to choose a subset of successes from a fixed number of trials. It is represented by combinations and is a critical component of the binomial probability formula, as it accounts for the different sequences in which successes can occur.

Transcript

00:01 Okay, so we're looking for the probability that this archer who hits his target 80 % of the time, so p equals 0 .8, and he shoots 7 arrows, so we have n equals 7.

00:15 And first off, we want where he never hits the target, so that's r equals 0.

00:21 So p r equals 0, which will be 7 to 0 .0.

00:32 We'll have our 0 .8 to the 0 with power.

00:38 0 .2 to the 7th power.

00:44 Then once we find what that probability is going to be, that'll be your answer, which i'll go ahead and write up at the end.

00:51 But we'll go on, and we'll talk about probability of b.

00:54 So probability to be that he hits the target each time, so he hits it every time.

00:58 So we'll do p of r equals 7.

01:02 Also, this is going to be equal to 1, because we'll have 7 factorial over 7 factorial.

01:07 And we'll have a very similar case here because we'll have 7 factorial over 7 factorial 0 factorial so this will be 1 and then we'll have 0 .8 to the 7th times 0 .2 to the 0th power and then for part c it says hits the target more than once so then i'm going to propose that the easiest way of doing this one so probability are greater than 1 would be just 1 minus minus p r equals 0 minus p of r equals 1, which we already have p of r equals 1, which so i guess i finally have to actually go ahead and figure out what that number is.

02:06 And that, okay, and so that number is going to be 0 .000128.

02:11 And also i went ahead and found our probability for r equals 7, which will be 0 .209752.

02:17 And now we'll want to find the probability of r equals 1 for this problem.

02:22 So our probability of r equals 1 is going to be the 7 factorial over 1 factorial over 6 factorial.

02:30 And then we'll have our 0 .8 to the 1 power times 0 .2 to the 6th power.

02:39 What that's going to end up giving us is going to be 7 times 0 .8 times 0 .2 to the 6th, which i'm getting to be about 0 .0003584.

02:57 And so when we do 1 minus that .003584 minus the .000128, we'll end up getting a very close number to just one.

03:17 So this is our answer right here actually...

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