An experiment consists of rolling a weighted die. The probability of rolling each number is: Pr[1]=0.05, Pr[2]=0.1, Pr[3]=0.25, Pr[4]=0.25, Pr[5]=0.2, and Pr[6]=0.15. On the first roll, you record if the number is Small (1,2) or Large (3,4,5,6). If the first number is Small, then on the second roll you record if the number is a 4 or not. If the first number is Large, on the second roll you record whether the number is Small or not. So, some typical outcomes would be S4 (small number rolled, then 4) or LL (large number rolled, then another large number). Draw a tree diagram and use it to answer the following questions.

Question

An experiment consists of rolling a weighted die. The probability of rolling each number is: Pr[1]=0.05, Pr[2]=0.1, Pr[3]=0.25, Pr[4]=0.25, Pr[5]=0.2, and Pr[6]=0.15. On the first roll, you record if the number is Small (1,2) or Large (3,4,5,6). If the first number is Small, then on the second roll you record if the number is a 4 or not. If the first number is Large, on the second roll you record whether the number is Small or not. So, some typical outcomes would be S4 (small number rolled, then 4) or LL (large number rolled, then another large number). Draw a tree diagram and use it to answer the following questions.

(1
What is Pr[(not4)onsecondroll|Sonfirstroll]?

(2) What is Pr[Sonsecondroll|Lonfirstroll]?

Answer 1

(1) Pr[(not 4) on second roll|S on first roll] can be obtained by looking at the branch that corresponds to getting a small number (S) on the first roll and then following the branch that leads to getting a number that is not 4 on the second roll.

Using the given probabilities, we have Pr[(not 4) on second roll|S on first roll]=0.9.

(2) Pr[S on second roll|L on first roll] can be obtained by looking at the branch that corresponds to getting a large number (L) on the first roll and then following the branch that leads to getting a small number on the second roll.

Using the given probabilities, we have Pr[S on second roll|L on first roll]=0.2.

Answer 2

To answer these questions, let's create a tree diagram to visualize the different outcomes.

Here is the tree diagram:

First Roll
/ \
S L
/ \ / \
4 not4 S L

Now, let's answer each question step-by-step:

(1) What is Pr[(not4)onsecondroll|Sonfirstroll]?
To calculate this probability, we need to focus on the path where the first roll results in a Small (S).
In this case, there are two possible outcomes: S4 and Snot4.

Probability of S4: Pr[S] * Pr[4|S] = 0.05 * 0.1 = 0.005
Probability of Snot4: Pr[S] * Pr[not4|S] = 0.05 * 0.9 = 0.045

Now, we can calculate the conditional probability:
Pr[(not4)onsecondroll|Sonfirstroll] = Probability of Snot4 / Probability of S
= 0.045 / 0.05
= 0.9

Therefore, Pr[(not4)onsecondroll|Sonfirstroll] is 0.9.

(2) What is Pr[Sonsecondroll|Lonfirstroll]?
To calculate this probability, we need to focus on the path where the first roll results in a Large (L).
In this case, there are two possible outcomes: LnotS and LL.

Probability of LnotS: Pr[L] * Pr[notS|L] = 0.95 * 0.75 = 0.7125
Probability of LL: Pr[L] * Pr[L|L] = 0.95 * 0.25 = 0.2375

Now, we can calculate the conditional probability:
Pr[Sonsecondroll|Lonfirstroll] = Probability of LL / Probability of L
= 0.2375 / 0.95
= 0.25

Therefore, Pr[Sonsecondroll|Lonfirstroll] is 0.25.

Answer 3

To answer these questions using a tree diagram, we can start by constructing the tree diagram representing the experiment described.

The first branch of the tree represents the first roll. We have two outcomes: Small (S) or Large (L).
- If the first roll is Small (S), we move to the second branch where we have two outcomes: 4 (4) or not 4 (not 4).
- If the first roll is Large (L), we move to the second branch where we have two outcomes: Small (S) or not Small (not S).

The tree diagram should look like this:

S
/ \
4 not 4

L
/ \
S not S

Now let's answer the questions using this tree diagram:

(1) Pr[(not 4) on second roll | S on first roll]:
To find this probability, we need to look at the branch where S is on the first roll. We follow that branch to the second roll and find the probability of landing on "not 4".

From the tree diagram, we see that the probability of landing on "not 4" given S on the first roll is 1, because once we have S on the first roll, we are guaranteed to move to the branch with "not 4" on the second roll.

Therefore, Pr[(not 4) on second roll | S on first roll] = 1.

(2) Pr[S on second roll | L on first roll]:
To find this probability, we need to look at the branch where L is on the first roll. We follow that branch to the second roll and find the probability of landing on S.

From the tree diagram, we see that the probability of landing on S given L on the first roll is 0.6 (Pr[3] + Pr[4] + Pr[5] + Pr[6] = 0.25 + 0.25 + 0.2 + 0.15 = 0.85, and Pr[3] + Pr[4] + Pr[5] + Pr[6] - Pr[4] = 0.85 - 0.25 = 0.6).

Therefore, Pr[S on second roll | L on first roll] = 0.6.

(1) Pr[(not 4) on second roll|S on first roll] can be obtained by looking at the branch that corresponds to getting a small number (S) on the first roll and then following the branch that leads to getting a number that is not 4 on the second roll.

To answer these questions, let's create a tree diagram to visualize the different outcomes.

To answer these questions using a tree diagram, we can start by constructing the tree diagram representing the experiment described.

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