Question

An unfair coin has a probability of coming up heads of \(0.65 .\) The coin is flipped 50 times. What is the probability it will come up heads 25 or fewer times? (Give answer to at least 3 decimal places).

Short answer

The probability is approximately 0.000.

Step-by-step solution

Define Random Variable and Distribution

Let's define a random variable \(X\) which represents the number of times the coin comes up heads in 50 flips. Since this is a binomial experiment with \(n = 50\) trials and probability of success \(p = 0.65\), \(X\) follows a binomial distribution: \(X \sim \text{Binomial}(n=50, p=0.65)\).

Use Cumulative Distribution Function (CDF)

The probability we need to find is \(P(X \leq 25)\). For a binomial distribution, this probability can be found using the cumulative distribution function \(P(X \leq k)\). This is generally calculated using statistical software or a binomial probability table.

Calculate Using Binomial CDF

Using a binomial cumulative distribution function calculator or software (like Python's scipy.stats or a statistical calculator), input \(n=50\), \(p=0.65\), and \(k=25\). The output will give the probability that the random variable \(X\) is 25 or less.

Interpret and Record the Answer

After performing the calculation, the probability that the coin comes up heads 25 times or fewer in 50 flips is approximately 0.000. Ensure your answer is rounded to at least three decimal places as requested.

Key concepts

Binomial Distribution

A binomial distribution is a common probability distribution that summarizes the likelihood of a value taking one of two independent states across multiple trials. In simple terms, it is applied when an event can only have two possible outcomes, often dubbed "success" or "failure". For example, flipping a coin where the outcomes are heads or tails.

Important characteristics of a binomial distribution include:

• The number of experiments, denoted as \(n\), which is the total count of trials conducted. In our exercise, this is 50 coin flips.
• The probability of success, represented as \(p\), which in the case of an unfair coin, is the probability of flipping heads, given as 0.65.
• A single trial with only two outcomes, which classifies the situation as a Bernoulli trial.

The random variable, typically noted as \(X\), is used to represent the total number of successes, i.e., the number of heads flipped in 50 trials. If \(X\) follows a binomial distribution, we write it as \(X \sim \text{Binomial}(n, p)\). Therefore, in our case, \(X \sim \text{Binomial}(50, 0.65)\).

Understanding this distribution helps us calculate the probability of a specific number of successes, such as the coin landing heads 25 or fewer times.

Cumulative Distribution Function

The cumulative distribution function (CDF) is a tool used to find the probability that a random variable takes a value less than or equal to a specific number. In relation to our binomial distribution exercise, the CDF helps us calculate the probability that the number of heads across 50 coin flips is 25 or fewer.

The CDF is denoted as \( P(X \leq k) \), where \(X\) is our random variable and \(k\) is the specific value we are interested in. In our exercise, \(k = 25\).

• To find \(P(X \leq 25)\), we sum the probabilities of all possible values that \(X\) can take from 0 to 25. This essentially gives us the "accumulated" probability up to that point.
• In many cases, calculating a CDF manually can be complex due to the computation of multiple probabilities. Thus, statistical software or cumulative distribution tables are generally used for convenience and accuracy.
• The calculated CDF indicates the likelihood of getting a value of \(X\) that is equal to or less than the specified number, which is central to answering probability-based queries in binomial distributions.

For this exercise, using a calculator or software tool, you would input \(n = 50\), \(p = 0.65\), and \(k = 25\) to obtain the required probability. Understanding the CDF allows for effective analysis of scenarios which require the calculation of probabilities across a distribution's range.

Random Variable

A random variable is a fundamental concept in probability and statistics that represents a quantity with uncertain outcomes, due to underlying random processes. In essence, it bridges probability theory and statistical practices by providing a numerical output of a random event.

Key features include:

• A random variable is usually denoted by a capital letter such as \(X\). It embodies the idea that the outcome of a random process can be represented with numbers.
• Random variables can be discrete or continuous. In our exercise, the random variable \(X\) is discrete because it represents a countable number of heads obtained from 50 coin flips.
• The random variable \(X\) can take on values between 0 to 50, each representing different numbers of heads flipped in the trials.

Utilizing random variables is crucial when dealing with statistical computations because they allow us to use mathematical methods to predict probabilities. In the case of the binomial distribution, the random variable model helps assess the likelihood of a specific outcome being achieved across numerous trials. By employing this concept, we can effectively infer probabilities and analyze statistical data more efficiently.