Let X be b(2, p) and let Y be b(4, p). If P(X ≥1)=(5)/(9), find P(Y ≥1).

Name: Problem 13, Chapter 3: Introduction to Mathematical Statistics
Uploaded: 2021-12-31T16:13:06-08:00
Duration: 6 min 54 s
Channel: Evelyn Cunningham
Description: Let X be b(2, p) and let Y be b(4, p). If P(X ≥1)=(5)/(9), find P(Y ≥1).

Let X be b(2, p) and let Y be b(4, p). If P(X ≥1)=(5)/(9),...

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials where each trial has the same probability of success. It is characterized by two parameters: the number of trials and the probability of success in each trial. The probability mass function is used to calculate the likelihood of obtaining a specific number of successes.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability that the event does not occur. This technique is particularly useful when calculating the probability of 'at least one' success, by subtracting the probability of zero successes from one.

Probability of At Least One Success

This concept refers to finding the chance that an event will occur at least once in a series of trials. It is frequently computed by using the complement rule on the probability of no occurrences, thereby simplifying the calculation in many applications.

Transcript

00:01 So we are going to let x be a binomial probability distribution with an n of 2 or a count of 2 and a probability of success p.

00:17 And we are going to let y be a binomial probability distribution with a count of 4 and the same probability of success.

00:30 So let's start with just drawing a picture of that.

00:36 So since x is a binomial distribution of count two, the values of the random variable could be a zero, a one, or a two, and for y, the random variables could be a zero, a one, a two, a three, or a four.

01:03 Now the second part of this, says if the probability of x being greater than or equal to one is five -ninths, then what is the probability of y being greater than or equal to one? so what we're going to do is we're going to start with this first probability distribution or of x.

01:29 And greater than or equal to one would be the numbers here.

01:36 And we know that these two total up to be five -ninths.

01:42 Now in any probability distribution, we know that the sum of all the probabilities of that distribution has to equal one.

01:51 So that means that the probability that x was equal to zero must be four ninths.

02:03 So we know that this has to be four nights.

02:07 We also know that the probability of x in a binomial probability has the formula ncx times p to the x times q to the n minus x.

02:25 Your textbook might also have it looking like this times p to the x times one minus p to the n minus.

02:36 And we already know that the probability of x being zero is four niths.

02:44 So if we know that the probability that x equals zero is two combination of two items taken zero at a time times p to the zero times 1 minus p to the 2 minus 0, just substituting values in.

03:12 And i know that the probability of x being 0 is 4 9ths.

03:17 Then i can substitute or replace the probability of x being 0 with 4 9ths.

03:22 And i'll have the combination of two items taken none at a time times p to the 0 times 1 minus p squared...

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