Question

Let \(x\) be a binomial random variable with \(n=7\) and \(p=.5 .\) Find the values of the quantities in Exercises \(11-15 .\) $$ P(x \leq 1) $$

Short answer

Answer: The probability is approximately 0.0625.

Step-by-step solution

Recall the binomial probability formula

The binomial probability formula is given by: $$ P(X=k) = \binom{n}{k} p^k (1-p)^{(n-k)} $$ Where \(n\) is the total number of trials, \(k\) is the number of successes, and \(p\) is the probability of success.

Calculate the probability for x = 0

We will use the binomial probability formula to calculate the probability of having 0 successes in 7 trials. In this case, n = 7, k = 0, and p = 0.5: $$ P(X=0) = \binom{7}{0} (0.5)^0 (1-0.5)^{(7-0)} = \frac{7!}{0!(7-0)!}(0.5)^7 = (1)(1)(0.5)^7 \approx 0.0078 $$

Calculate the probability for x = 1

Now, we will calculate the probability of having 1 success in 7 trials. In this case, n = 7, k = 1, and p = 0.5: $$ P(X=1) = \binom{7}{1} (0.5)^1 (1-0.5)^{(7-1)} = \frac{7!}{1!(7-1)!}(0.5)^7 = (7)(0.5)(0.5)^6 \approx 0.0547 $$

Calculate the probability \(P(x \leq 1)\)

Finally, to find the probability \(P(x \leq 1)\), we will add the probabilities calculated in Steps 2 and 3: $$ P(x \leq 1) = P(X=0) + P(X=1) \approx 0.0078 + 0.0547 \approx 0.0625 $$ So, the probability of having at most 1 success in 7 trials is approximately 0.0625.

Key concepts

Binomial Random Variable

In probability theory and statistics, a binomial random variable is a specific type of discrete random variable. It counts how often a particular event, usually referred to as a 'success,' occurs in a fixed number of trials or experiments.

Each trial is independent, meaning the outcome of one trial does not affect another, and there are only two possible outcomes: success or failure. The two main parameters that define a binomial random variable are the number of trials, denoted by the letter 'n', and the probability of success in each trial, denoted by the letter 'p'. For instance, in a coin toss, if we define 'success' as landing heads, and we flip the coin 7 times, this scenario can be described by a binomial random variable.

The formula to find the probability of obtaining exactly 'k' successes in 'n' trials is given by the binomial probability formula. As applied to our coin example, to find the probability of achieving exactly one head (success) in seven tosses (trials), we would use this formula to perform the calculations.

Binomial Distribution

A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions. Essentially, it reflects the distribution of a binomial random variable.

Visualizing this distribution, we often use a histogram or a probability mass function plot, where each bar represents the probability of obtaining each possible number of successes (from 0 up to 'n'). The shape of the distribution is determined by the parameters 'n' and 'p'. As 'n' increases or as 'p' gets closer to 0.5, the distribution tends to look more symmetrical.

For practical applications, knowing the binomial distribution helps in understanding the variability and possible outcomes of processes that follow a binomial pattern - for example, quality control testing in manufacturing, where 'success' could be a product passing inspection.

Probability of Success

In the context of binomial variables and distribution, the probability of success 'p' is a crucial concept. It is the likelihood that any given trial will result in a success. This probability is assumed to be constant for each trial within the series.

If we return to the coin example, assuming a fair coin, the probability of getting heads (our defined 'success') is 0.5, because there is an equal chance of landing heads or tails. The 'probability of success' directly affects the possible outcomes, and it is essential to accurately determine this probability when analyzing real-world situations.

To calculate cumulative probabilities, like 'What is the probability of achieving one or fewer heads in seven coin tosses?', you sum the individual probabilities of all events from 0 up to your desired number of successes, computing each by employing the binomial formula with our known 'n' and 'p'. Depending on the values of 'n' and 'p', these cumulative probabilities can offer significant insights into the likelihood of a range of outcomes and are highly informative for predicting and understanding the behavior of binomial processes.