Question

The following questions are from ARTIST (reproduced with permission) Five faces of a fair die are painted black, and one face is painted white. The die is rolled six times. Which of the following results is more likely? a. Black side up on five of the rolls; white side up on the other roll b. Black side up on all six rolls c. a and b are equally likely

Short answer

Option a is more likely.

Step-by-step solution

Understand the Probability Concept

We are dealing with a die that has six faces: five black and one white. Each roll is an independent event, with the probability being consistent, as it is a fair die. The probability of rolling a black face is \( \frac{5}{6} \) and the probability of rolling a white face is \( \frac{1}{6} \).

Calculate Probability for Scenario a

We need to calculate the probability that the black side is rolled five times and the white side once. This is a binomial probability problem. The formula for binomial probability is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]For scenario a, \( n = 6 \), \( k = 5 \), \( p = \frac{5}{6} \), \( 1-p = \frac{1}{6} \). Thus:\[P(a) = \binom{6}{5} \left(\frac{5}{6}\right)^5 \left(\frac{1}{6}\right)^{1}\]Calculate \( \binom{6}{5} \), which is 6, and then \( \left(\frac{5}{6}\right)^5 \) and \( \left(\frac{1}{6}\right) \).

Calculate Probability for Scenario b

For scenario b, we want all six rolls to show a black side, using the same binomial formula with \( k = 6 \):\[P(b) = \binom{6}{6} \left(\frac{5}{6}\right)^6 (\frac{1}{6})^0\]Since \( \binom{6}{6} = 1 \), it simplifies to:\[P(b) = \left(\frac{5}{6}\right)^6\]Calculate \( \left(\frac{5}{6}\right)^6 \).

Compare the Probabilities

Now, we compare the probabilities calculated in steps 2 and 3. From step 2:\[P(a) = 6 \times \left(\frac{5}{6}\right)^5 \times \left(\frac{1}{6}\right)\]From step 3:\[P(b) = \left(\frac{5}{6}\right)^6\]Determine which of these two probabilities is greater. If \( P(a) > P(b) \), scenario a is more likely. If they are equal, then c is correct.

Key concepts

Binomial Distribution

A binomial distribution is a probability distribution that summarizes the likelihood of obtaining a fixed number of successful outcomes in a specific number of trials. This is an essential concept in probability theory when dealing with events that have two distinct results, like "success" and "failure." In our problem with the die, the successful outcome can be rolling a black face, while the failure is rolling a white face.

For our fair die, we're rolling it six times. Here, each roll represents a single trial. Out of six possible outcomes, the success is rolling a black face, which occurs with a probability of \( \frac{5}{6} \), while the failure, rolling a white face, happens with probability \( \frac{1}{6} \).

• The formula for binomial probability is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\]
• Here, \( n \) is the total number of trials (rolls); \( k \) is the number of successful outcomes; and \( p \) is the probability of success for each trial.

This formula helps determine the likelihood across different scenarios, such as five black faces and one white face, or six black faces in our exercise.

Independent Events

In probability theory, independent events are events whose outcomes do not affect one another. Each roll of a die is typically considered independent because the result of one roll does not influence the next. In our exercise, every time we roll the die, it has no memory of the previous outcomes.

This concept is crucial for calculating probabilities over multiple trials. When dealing with things like dice, cards, or coins, understanding independence means you can multiply probabilities across trials to find combined likelihoods.

• For instance, the probability of rolling a black face on two consecutive rolls in our scenario can be found by: \( \frac{5}{6} \times \frac{5}{6} \).
• Independence simplifies calculation across multiple trials, allowing the application of probability rules with straightforward multiplication.

Knowing that the events are independent helps in applying the binomial formula correctly as it counts these trials separately without effect on each other.

Fair Die

A fair die is an important object in probability because it ensures equal likelihood for each face to appear when rolled. In this exercise, we are working with a slightly different scenario – a modified die where five faces are black, and one face is white.

Despite this uneven color distribution, each face still has a defined probability of appearing under a single roll (though it's not 1 out of 6 as in a truly fair die typically would be). This configuration adds layers to our probability calculations.

• The probability of rolling a black face is \( \frac{5}{6} \) with our modified fair die.
• The probability for a white face is \( \frac{1}{6} \).

Understanding these probabilities is key to properly interpreting the chances of different scenarios in our original problem. Even though the dice are fair mechanically (each face is equally weighted), the unequal coloring provides a non-standard probability distribution critical for the given task.