Finding Binomial Probabilities-Excel: Videos & Practice Problems

Calculating binomial probabilities can be simplified significantly by using Excel's BINOM.DIST function, which streamlines the process compared to manual calculations with the binomial probability formula. This function requires four key arguments: the number of successes (x), the number of trials (n), the probability of success in each trial (p), and a logical value indicating whether to calculate the exact probability or the cumulative probability.

For example, consider a scenario where a spam filter correctly identifies spam emails with a 92% success rate. If an employee receives 361 spam emails in a year, you might want to find the probability that exactly 320 emails are correctly flagged. Here, the number of successes x is 320, the number of trials n is 361, and the probability of success p is 0.92 (expressed as a decimal). Using the BINOM.DIST function in Excel, you input these values as =BINOM.DIST(320, 361, 0.92, FALSE) to find the exact probability that exactly 320 emails are correctly identified.

When interested in the probability that no more than 320 emails are correctly identified (i.e., the cumulative probability where x is less than or equal to 320), the same function is used but with the last argument set to TRUE: =BINOM.DIST(320, 361, 0.92, TRUE). This returns the cumulative probability, which sums the probabilities of all outcomes from 0 up to 320 successes.

To find the probability that more than 320 emails are correctly identified, Excel's BINOM.DIST function cannot directly calculate this because it only handles exact or cumulative probabilities up to a value. Instead, the complement rule is applied: the probability that x is greater than 320 equals one minus the cumulative probability that x is less than or equal to 320. In Excel, this is expressed as =1 - BINOM.DIST(320, 361, 0.92, TRUE). This approach leverages the fundamental probability principle that the sum of complementary events equals one.

Using Excel functions like BINOM.DIST not only saves time but also reduces errors in calculating binomial probabilities, especially for large numbers of trials. Understanding how to set the arguments correctly and apply the complement rule enhances your ability to analyze binomial distributions effectively, whether for exact outcomes or cumulative probabilities.

In analyzing the performance of an ice hockey player who scores on 6.4% of their shots, we consider the total number of shots taken in a season, which is 211. The problem involves calculating various probabilities related to the number of goals scored, modeled as a binomial distribution where each shot is a trial with a success probability p = 0.064.

To find the probability of scoring exactly 20 goals, we use the binomial probability formula, which calculates the likelihood of x successes in n independent trials with success probability p. The key parameters are x = 20, n = 211, and p = 0.064. The binomial probability mass function (PMF) is given by:

Using computational tools, this probability is approximately 0.02, indicating a 2% chance the player scores exactly 20 goals in the season.

Next, to determine the probability that the player scores no more than 20 goals (i.e., \( P(X \leq 20) \)), we use the cumulative distribution function (CDF) of the binomial distribution. This sums the probabilities of scoring 0 up to 20 goals. The cumulative probability is about 0.97, meaning there is a 97% chance the player scores 20 or fewer goals.

For the probability of scoring more than 20 goals (\( P(X > 20) \)), we apply the complement rule, recognizing that the events \( X \leq 20 \) and \( X > 20 \) are mutually exclusive and exhaustive. Thus:

Given the cumulative probability of 0.97, the probability of scoring more than 20 goals is approximately 0.03, or 3%.

To find the probability of scoring less than 20 goals (\( P(X < 20) \)), we subtract the probability of scoring exactly 20 goals from the cumulative probability of scoring no more than 20 goals:

This results in about a 95% chance that the player scores fewer than 20 goals.

Finally, to calculate the probability that the player scores between 20 and 25 goals inclusive (\( 20 \leq X \leq 25 \)), we find the cumulative probability up to 25 goals and subtract the cumulative probability of scoring less than 20 goals:

Using the binomial CDF for \( P(X \leq 25) \) and the previously calculated \( P(X < 20) \), this probability is approximately 0.05, or 5%. This approach highlights the utility of the binomial distribution and complement rule in calculating exact and cumulative probabilities for discrete random variables, especially in sports analytics contexts.