Chapter 3: 3.2 (page 97)

If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is $i$ ? Compute for all values of $i$ between $2$ and $12$

Short Answer

Expert verified

$\begin{aligned} P (A ∣ i = 2) = 0 \\ P (A ∣ i = 3) = 0 \\ P (A ∣ i = 4) = 0 \\ P (A ∣ i = 5) = 0 \\ P (A ∣ i = 6) = 0 \\ P (A ∣ i = 7) = \frac{1}{6} \\ P (A ∣ i = 8) = \frac{1}{5} \\ P (A ∣ i = 9) = \frac{1}{4} \\ P (A ∣ i = 10) = \frac{1}{3} \\ P (A ∣ i = 11) = \frac{1}{2} \\ P (A ∣ i = 12) = 1 \end{aligned}$

Step by step solution

Given Information

The number on first dice= $6$

Sum of both dice= $i$ .

Calculation.

When i = $2, 3, 4, 5, 6$

Consider that $A$ is the event that first lands in $6$ .

$\begin{aligned} P (A ∣ i = 2) = 0 \\ P (A ∣ i = 3) = 0 \\ P (A ∣ i = 4) = 0 \\ P (A ∣ i = 5) = 0 \\ P (A ∣ i = 6) = 0 . \end{aligned}$

$\begin{aligned} P (A ∣ i = 7) = \frac{1}{6} \end{aligned}$

$\begin{aligned} P (A ∣ i = 8), that is i = 8 occurs when (2, 6) (6, 2) (5, 3) (3, 5) (4, 4) \end{aligned}$

Thus,

$P (A ∣ i = 9), that is i = 9 occurs when (3, 6) (6, 3) (5, 4) (4, 5)$

$\begin{aligned} P (A ∣ i = 8) = \frac{1}{5} \\ P (A ∣ i = 9), that \\ Thus, \\ P (A ∣ i = 9) = \frac{1}{4} \end{aligned}$

$P (A ∣ i = 10)$ that is $i = 10$ occurs when $(4, 6) (6, 4) (5, 5)$

Thus,

$P (A ∣ i = 10) = \frac{1}{3}$

$P (A ∣ i = 11)$ that is $i = 11$ occurs when $(5, 6) (6, 5)$

$P (A ∣ i = 11) = \frac{1}{2}$

$P (A ∣ i = 12)$ , that is $i = 12$ occurs when $(6, 6)$

Thus,

$P (A ∣ i = 12) = 1$

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If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is $i$ ? Compute for all values of $i$ between $2$ and $12$

Short Answer

Step by step solution

Given Information

Calculation.

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If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is i? Compute for all values of ibetween 2and12

Short Answer

Step by step solution

Given Information

Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Discrete Mathematics

Calculus

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

If two fair dice are rolled, what is the conditional probability that the first one lands on 6 given that the sum of the dice is $i$ ? Compute for all values of $i$ between $2$ and $12$