In Exercises 15 and 16, describe and correct the error in...

In Exercises 15 and $16,$ describe and correct the error in calculating the probability of rolling a 1 exactly 3 times in 5 rolls of a six-sided die.
$$\begin{aligned} P(k=3) &=\left(\frac{1}{6}\right)^{3}\left(\frac{5}{6}\right)^{5-3} \\ & \approx 0.003 \end{aligned}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Probability of Success and Failure

In a binomial setting, each trial has a probability of success and a probability of failure, which are raised to the power corresponding to the number of successes and failures, respectively. These probabilities must be multiplied by the combination factor to accurately reflect the overall probability of a given outcome.

Combination Factor (n choose k)

The combination factor, typically represented as 'n choose k', indicates the number of ways to select k successes out of n trials without considering the order of occurrence. In probability problems using the binomial distribution, omitting this factor leads to an incomplete calculation, as it accounts for the different arrangements in which the successes can occur.

Binomial Probability Distribution

This distribution is used to determine the likelihood of obtaining a fixed number of successes in a set number of independent trials, each with exactly two outcomes (success or failure). It is fundamental in calculating probabilities where each trial has the same chance of success, such as rolling a die with a specific outcome.

Transcript

Please give Ace some feedback