Question

Assume that \(13 \%\) of people are left-handed. If we select 5 people at random, find the probability of each outcome described below. a) The first lefty is the fifth person chosen. b) There are some lefties among the 5 people. c) The first lefty is the second or third person. d) There are exactly 3 lefties in the group. e) There are at least 3 lefties in the group. f) There are no more than 3 lefties in the group.

Short answer

a) 0.068, b) 0.499, c) 0.116, d) 0.035, e) 0.039, f) 0.996.

Step-by-step solution

Define Probability Values

Let the probability of selecting a left-handed person be \( p = 0.13 \), and the probability of selecting a right-handed person be \( q = 1-p = 0.87 \).

Part a - Finding Probability for First Lefty Being Fifth

To find the probability that the first lefty is the fifth person, we need four right-handed people first and a lefty last. This probability is \( q^4 \times p = 0.87^4 \times 0.13 = 0.068 \).

Part b - Some Lefties Among 5 People

The probability of having some lefties is the complement of having no lefties. Calculate the probability of no lefties and subtract from 1: \( 1 - q^5 = 1 - 0.87^5 = 0.499 \).

Part c - The First Lefty Being Second or Third

Compute the probability for the lefty being the second or the third person. For the second: \( q \times p \times q^3 \). For the third: \( q^2 \times p \times q^2 \). Sum them: \( 0.87 \times 0.13 \times 0.87^3 + 0.87^2 \times 0.13 \times 0.87^2 = 0.116 \).

Part d - Exactly 3 Lefties

The probability of exactly 3 lefties follows binomial distribution: \( \binom{5}{3} \times p^3 \times q^2 = 10 \times (0.13)^3 \times (0.87)^2 = 0.035 \).

Part e - At Least 3 Lefties

Add probabilities for cases with 3, 4, or 5 lefties. Calculate: \( \binom{5}{3} \times p^3 \times q^2 + \binom{5}{4} \times p^4 \times q^1 + \binom{5}{5} \times p^5 \).Calculate individually and sum: \( 0.035 + 0.004 + 0.00003 = 0.039 \).

Part f - No More Than 3 Lefties

The probability of no more than 3 lefties is the complement of having more than 3: \( 1 - (\text{probability of 4 or 5 lefties}) \). We found 4 and 5 together as 0.00403. So, \( 1 - 0.00403 = 0.996 \).

Key concepts

Left-handed Probability

When considering the probability of left-handed individuals in any group, it's important to understand the basic probability rules that apply.
Given that the proportion of left-handed people is approximately 13%, which we represent as the probability of selecting a left-handed person, denoted by \( p = 0.13 \). For right-handed people, the probability is \( q = 1 - p = 0.87 \). This means that when randomly selecting individuals, there's a consistent chance that each one can either be left-handed or right-handed.
In a variety of scenarios — such as finding the first left-handed person in a sequence or determining how many left-handed people are in a group — it's essential to consider this base probability. Each situation requires calculating probabilities based on these fundamental values, often using combinations or specific sequences of events.

Probability Distributions

Probability distributions provide a way to model and predict different outcomes based on probability. In our exercise, the binomial distribution is particularly useful.
This type of distribution helps us calculate the likelihood of a specific number of successes (in our case, selecting lefties) in a fixed number of trials (like picking 5 people).
To find the probability of getting exactly 3 left-handed people in a group of 5, we use the formula for the binomial distribution:

• The formula is \( \binom{n}{k} \times p^k \times q^{n-k} \), where \( n \) is the total number of trials, \( k \) is the number of successful trials (lefties), \( p \) is the probability of success, and \( q \) is the probability of failure.
• In our example, each calculation uses \( n = 5 \), \( p = 0.13 \), and \( q = 0.87 \).

This distribution helps us to compute probabilities not only for fixed outcomes but also for ranges, like calculating probabilities for at least or at most a certain number of left-handed people.

Statistical Calculations

Statistical calculations in this context involve applying the right probability formulas to obtain desired outcomes.
For example, the calculation in part (b) determines the probability of having 'some lefties' out of 5 people. To achieve this, you first find the probability of having no lefties at all (which is easier to calculate) and subtract it from 1. This method is known as finding the complement.
These statistical calculations involve both simple operations, like multiplying probabilities, and more complex ones like summing or subtracting probabilities of several independent events. Understanding these methods can help tackle real-world problems that deal with statistical likelihood, allowing for an informed approach to estimating outcomes based on current known probabilities.