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Chapter 7: Problem 34

Find each of the following probabilities when n independent Bernoulli trials are carried out with probability of success p. a) the probability of no successes b) the probability of at least one success c) the probability of at most one success d) the probability of at least two successes

Short Answer

Expert verified
a) \( (1 - p)^n \). b) \( 1 - (1 - p)^n \). c) \[ (1 - p)^n + n \times p \times (1 - p)^{n-1} \]. d) \[ 1 - \text{( (1 - p)^n + n \times p \times (1 - p)^{n-1} )} \].

Step by step solution

01

Understand the Bernoulli Trial

In a Bernoulli trial, there are only two possible outcomes: success or failure. Each trial is independent of the others, and the probability of success is denoted by \( p \).
02

Calculate Probability of No Successes

The probability of no successes in \( n \) independent Bernoulli trials is given by \( (1 - p)^n \). This is because each trial must result in a failure, which has a probability of \( 1 - p \). Therefore, the probability of all \( n \) trials failing is \( (1 - p)^n \).
03

Calculate Probability of At Least One Success

To find the probability of at least one success, use the complement rule. The probability of at least one success is \( 1 - \text{(probability of no successes)} \), which is \( 1 - (1 - p)^n \).
04

Calculate Probability of At Most One Success

For at most one success, combine the probabilities of no successes and exactly one success. The probability of exactly one success is \( n \times p \times (1 - p)^{n-1} \). Therefore, the probability of at most one success is \[ (1 - p)^n + n \times p \times (1 - p)^{n-1} \].
05

Calculate Probability of At Least Two Successes

Use the complement rule again for finding the probability of at least two successes. This is \[ 1 - \text{(probability of zero successes + probability of exactly one success)} \], which simplifies to \[ 1 - ( (1 - p)^n + n \times p \times (1 - p)^{n-1} ) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability of No Successes
In a series of Bernoulli trials, you either have a success or a failure for each trial. The probability of not having any success in a set of n independent trials, each with a success probability of p, is found by considering all trials failing.
The formula to calculate this is: \( (1 - p)^n \).
This comes from the fact that each trial has a failure probability of \( 1 - p \). When every trial results in failure, the overall probability is given by raising \( (1 - p) \) to the power of \( n \).
For example, if you conduct 4 trials, and the probability of success in each trial is 0.3, the probability of no successes would be: \( (1 - 0.3)^4 \).
The result will give you an idea of how likely it is to not have any successes at all!
Probability of At Least One Success
To find the probability of having at least one success in n Bernoulli trials, use the complement rule. This means you calculate the probability of the opposite event (getting no successes at all) and subtract it from 1.
The formula is: \( 1 - (1 - p)^n \).
This uses the idea that it's easier to first find the probability of zero successes and then subtract from the total probability of all events which is 1.
For instance, if you perform 5 trials with a success chance of 0.4 for each trial, the probability of at least one success is: \( 1 - (1 - 0.4)^5 \).
Simply put, start by figuring out the chance of getting no success and then subtract this value from 1.
Probability of At Most One Success
Sometimes you're interested in the likelihood of getting no more than one success from n trials. For this, combine the probabilities of getting zero successes and exactly one success.
First, zero successes is \( (1 - p)^n \).
Next, for exactly one success: one trial succeeds, and the rest fail. The formula is: \( n \times p \times (1 - p)^{n-1} \).
Thus, the combined probability for at most one success is:\( (1 - p)^n + n \times p \times (1 - p)^{n-1} \).
For example, with 3 trials and a success probability of 0.5 for each trial, the probability of getting at most one success can be found using the above formulas.
This approach gives you a full picture of the chances of having zero or one successes in a set of trials.
Probability of At Least Two Successes
To determine the probability of getting at least two successes, again use the complement rule, where you consider the easier-to-calculate events and subtract from 1.
This involves subtracting the probabilities of getting zero and exactly one success from 1.
The formula to use is:\( 1 - ( (1 - p)^n + n \times p \times (1 - p)^{n-1} ) \).
By doing this, you're excluding the cases where there are no successes or just one success, leaving only the scenarios with at least two successes.
Let鈥檚 say you perform 4 trials, each with a success probability of 0.2. Plug the values into the formula to find the probability of getting at least two successes.
This method simplifies the process by focusing on the complementary events first, making calculations more straightforward.

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Most popular questions from this chapter

A coin is biased so that the probability a head comes up when it is flipped is 0.6. What is the expected number of heads that come up when it is flipped 10 times?

A pair of dice is rolled in a remote location and when you ask an honest observer whether at least one die came up six, this honest observer answers in the affirmative. a) What is the probability that the sum of the numbers that came up on the two dice is seven, given the information provided by the honest observer? b) Suppose that the honest observer tells us that at least one die came up five. What is the probability the sum of the numbers that came up on the dice is seven, given this information?

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