Problem 45 - Algebra for College Students

Question

Four coins are tossed 144 times. How many times would you expect to get four tails? 9

Step-by-Step Solution

Answer

You expect to get four tails 9 times.

Step 1: Understanding the Problem

We need to find out how many times we expect to get four tails when four coins are tossed 144 times. This is a problem involving probability and expected value.

Step 2: Finding the Probability

First, calculate the probability of getting tails on a single coin toss, which is \( \frac{1}{2} \). Since we need four tails, the probability of this event occurring in one toss of four coins is \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \).

Step 3: Calculating Expected Value

The expected number of times to get four tails in 144 tosses can be found by multiplying the probability of getting four tails by the number of trials. Thus, it is \( 144 \times \frac{1}{16} = 9 \).

Step 4: Final Calculation Verification

Verify the calculation for accuracy. \( 144 \times \frac{1}{16} = 144 \div 16 = 9 \). This confirms the expected number of times to get four tails is indeed 9.

Key Concepts

Understanding Expected Value

The expected value is a fundamental concept in probability and statistics. It refers to the average or mean outcome you would anticipate over a large number of trials of a probabilistic event. In simpler terms, it's what you would "expect" to happen on average.
The expected value helps us in decision-making by providing a long-term perspective on the likely outcomes of different events. To compute it, we multiply each potential outcome by its probability and sum those products.

• Expected value = probability of an outcome × number of trials

In the context of our problem, the expected value tells us how many times we can expect to achieve four tails when tossing four coins 144 times. The calculation involves knowing the probability of the desired outcome (four tails) and then considering how often we perform the experiment (144 trials). This allows us to predict the number of successes over the course of many repetitions.

Decoding Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood of a particular number of successes in a specific number of trials of a binary experiment. A binary experiment is one in which there are only two possible outcomes, like flipping a coin.
The binomial distribution is governed by two parameters:

• n: the number of trials
• p: the probability of success on an individual trial

In our coin tossing problem, each toss of four coins and getting four tails can be considered a success, with n being the total number of toss events and p being the probability of achieving the desired outcome in those events.
This distribution is crucial in calculating expected values because it allows the determination of the probability of a certain number of successes out of many attempts. The binomial probability formula is given by \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \) where \( P(X = k) \) is the probability of getting k successes in n trials. In simpler coin toss scenarios, like our problem, direct calculation of expected outcomes is more practical.

Exploring Coin Toss Experiments

Coin toss experiments are classic examples in probability theory as they distinctly illustrate the concepts of binary outcomes (heads or tails) and randomness.
The simplicity of a coin toss makes it a great introduction to probability concepts such as independence and fair chance. Each toss of a fair coin is an independent event, meaning the outcome of one toss does not influence the next.

• Probability of heads: \( \frac{1}{2} \)
• Probability of tails: \( \frac{1}{2} \)

In our problem, tossing four coins aims to get four tails, which is an uncommon event considering each coin behaves independently. The probability of getting tails on any single toss is \( \frac{1}{2} \), so for all four coins to land tails, the probability becomes \( \left(\frac{1}{2}\right)^4 = \frac{1}{16} \). This is why the occurrence of four tails in numerous trials is infrequent and requires careful probability calculation.