An experiment consists of rolling a weighted die. The probability of rolling each number is: Pr[1]=0.2 , Pr[2]=0.15 , Pr[3]=0.25 , Pr[4]=0.15 , Pr[5]=0.05 , and Pr[6]=0.2 . 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.
What is Pr[LL]?
What is Pr[(not4)on second roll/S on first roll]?
What is Pr[S on second roll/L on first roll]
To draw a tree diagram for this experiment, we start with two branches representing the two possibilities on the first roll: Small (S) and Large (L).
For the Small branch, we have another branch representing the second roll outcomes: 4 and not4.
For the Large branch, we have another branch representing the second roll outcomes: Small and notSmall.
Here is the tree diagram:
Start
/ \
S L
/ \ / \
4 not4 S L
Now, let's answer the questions using the tree diagram:
1. Pr[LL]:
To find the probability of rolling LL, we follow the path through the tree diagram. It is the probability of rolling a Large (L) on the first roll and then another Large (L) on the second roll. Looking at the diagram, we can see that the probability of rolling L on the first roll is 0.55 (0.25 + 0.15 + 0.05 + 0.2), and the probability of rolling L on the second roll, given that L was rolled on the first roll, is 0.2. Therefore, the probability of LL is 0.55 * 0.2 = 0.11.
So, Pr[LL] = 0.11.
2. Pr[(not4) on second roll / S on first roll]:
To find the probability of rolling (not 4) on the second roll given that S was rolled on the first roll, we look at the path S -> not4 in the tree diagram. The probability of rolling S on the first roll is 0.35 (0.2 + 0.15) and the probability of rolling not4 on the second roll, given that S was rolled on the first roll, is 0.83 (1 - 0.17, where 0.17 is the probability of rolling 4 on the second roll). Therefore, the probability of (not 4) on the second roll given S on the first roll is 0.35 * 0.83 = 0.2905.
So, Pr[(not4) on second roll / S on first roll] = 0.2905.
3. Pr[S on second roll / L on first roll]:
To find the probability of rolling S on the second roll given that L was rolled on the first roll, we look at the path L -> S in the tree diagram. The probability of rolling L on the first roll is 0.55 (0.25 + 0.15 + 0.05 + 0.2) and the probability of rolling S on the second roll, given that L was rolled on the first roll, is 0.36 (0.2 + 0.16). Therefore, the probability of S on the second roll given L on the first roll is 0.55 * 0.36 = 0.198.
So, Pr[S on second roll / L on first roll] = 0.198.