Question

Let an unbiased die be cast at random seven independent times. Compute the conditional probability that each side appears at least once given that side 1 appears exactly twice.

Short answer

The conditional probability that each side appears at least once given that side 1 appears exactly twice is \( \frac{2 \times \binom{5}{1} \times \binom{5}{2} \times 4!}{\binom{7}{2} \times 5^5} \).

Step-by-step solution

Determine the Condition - Side 1 Appears Exactly Twice

Considering the given condition, side 1 can appear twice in \( \binom{7}{2} = 21 \) ways out of the total \( 6^7 \) possible throws.

Find the Outcomes where Each Side Appears At Least Once

To find the outcomes where each side appears at least once, we first calculate all possible combinations of the outcomes: \( 6^7 \). Then subtract combinations where at least one number is absent: \( 6 \times 5^7 - \binom{6}{2} \times 4^7 + \binom{6}{3} \times 3^7 - \binom{6}{4} \times 2^7 + \binom{6}{5} \) (inclusion-exclusion principle).

Calculate The Conditional Probability

The conditional probability that each side appears at least once given that side 1 appears exactly twice is calculated by dividing the number of ways where each side appears at least once and exactly 2 ones occur: \( 2 \times \binom{5}{1} \times \binom{5}{2} \times 4! \), by the total number of ways where side 1 appears twice: \( \binom{7}{2} \times 5^5 \). Hence, \( P = \frac{2 \times \binom{5}{1} \times \binom{5}{2} \times 4!}{\binom{7}{2} \times 5^5} \).

Key concepts

Probability Theory

Probability theory is the mathematical framework for quantifying the likelihood of certain events occurring. In the context of rolling an unbiased die, which has six faces, the probability of any specific outcome is equally likely. For example, the probability of rolling a four in a single throw is one in six, or roughly 16.67%.

When discussing conditional probability, as in the exercise, we are focusing on a specific situation where the probability of an event is calculated under the condition that another event has occurred. For example, the question asks us to determine the likelihood of all six sides appearing at least once, given that one specific side (side 1) appears exactly twice in seven throws. This concept allows us to adjust our expectations based on new information, such as the appearance of side 1 exactly twice, which creates a more refined and accurate assessment of the situation.

Combinatorics

Combinatorics is the area of mathematics dedicated to counting, arranging, and finding patterns within discrete structures. It is key to solving many probability problems, like the one involving dice. The use of combinatorial calculations, such as combinations and permutations, can elucidate the number of possible outcomes in a given scenario.

In our dice-throwing exercise, combinatorics is utilized to count how many ways side 1 can appear exactly twice in seven throws. The notation \( \binom{n}{k} \) represents the combination, which tells us how many ways we can choose \(k\) items from \(n\) without considering the order. Here, it's used to calculate the number of possible sequences where two throws result in side 1, and the order in which these occur does not matter. Understanding combinatorics is essential to breaking down complex probability questions into manageable parts that can be counted and consequently evaluated.

Inclusion-Exclusion Principle

The inclusion-exclusion principle is an important concept in combinatorics and probability that deals with the counting of elements in overlapping sets. It provides a way to calculate the total number of elements in a finite number of sets, including cases where sets intersect, by adding the sizes of the individual sets and then subtracting the sizes of the pairwise intersections, adding back the size of triple intersections, and so on.

In the die-throwing example, this principle helps us subtract combinations where at least one number doesn't appear, thereby correcting for overcounting. To apply the principle, you begin with the total number of outcomes and subtract out the sets of throws where one or more numbers are missing. This correction is essential in order to accurately determine the number of valid outcomes where all sides appear at least once. Understanding the principle is crucial for problems where multiple conditions can overlap, and it helps prevent miscounting those overlaps when assessing the probability of certain complex events.