An Olympic archer is able to hit the bull's-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.

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.

Key Concepts

Bernoulli Trials

Bernoulli trials are experiments or processes in which there are exactly two possible outcomes (commonly termed as 'success' and 'failure') for each individual trial. Each trial is independent of the others, and the probability of success remains constant. This forms the basis for many discrete probability distributions when dealing with repeated experiments.

Geometric Distribution

The geometric distribution models the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials. It is particularly useful when calculating probabilities such as 'the first success occurs on the nth trial', and its probability mass function reflects the need for a series of failures preceding a success.

Binomial Distribution

The binomial distribution describes the probability of obtaining a fixed number of successes in a specified number of independent Bernoulli trials. It uses the combination formula to account for the different ways successes can occur among the trials, and it is essential for calculating probabilities such as exactly k successes, at least k successes, or at most k successes.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of the event not occurring. This rule is particularly useful for simplifying probability calculations, such as determining the likelihood of at least one success by subtracting the probability of zero successes from one.

Cumulative Distribution

The cumulative distribution function (CDF) provides the probability that a random variable takes on a value less than or equal to a certain threshold. This concept is helpful when calculating probabilities like 'at least' or 'at most' a certain number of successes, as it allows one to sum the appropriate probabilities or use complementary logic.

Transcript

00:01 For this exercise, we are told that an olympic archer is able to hit the bullseye 80 % of the time, and that each shot is independent of the others.

00:09 So they're telling us that these are bernoulli trials.

00:12 So each trial has success or failure outcomes, and each trial is independent from the others.

00:22 So we're told that she's going to shoot six arrows, and we answer some probability questions.

00:31 For part a, we're asked, what is the probability that her first bullseye comes on the third arrow.

00:40 So here we can define x as the number of trials until the first success.

00:55 And in this case, x is a geometric random variable with a probability of 0 .8.

01:10 So we'll call that a success hitting a bullseye.

01:18 So the probability of her first success coming on the third trial is the probability that x is equal to 3.

01:27 And you may recall that generally, the probability of x equaling k for a geometric random variable is equal to q to the exponent k minus 1 times p so here we have 0 .2 to the exponent 2 times 0 .8 and this comes out to 0 .032.

02:02 So there's about a 3 .2 % chance that her first bullseye comes on the third arrow for part b we are asked the probability that she misses the bullseye at least once.

02:18 So for this type of question, we want to define the random variable as the number of successes in the six trials.

02:36 And this type of random variable is the binomial random variable.

02:41 And this one is based on six trials and probability of success of 0 .8.

02:49 So if we want the probability that she misses the bullseye at least once, that's the same as the probability...

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