your teacher has invented a fair dice game to play. your teacher will roll one fair eight sided die and you will roll a fair six sided die. each player rolls once and the winner is the person with the higher number. in case of a tie neither player wins.
a. let A be the event "your teacher wins" find P(A).
B. let b be the event "you get a 3 on your first roll.+ Find P( A u B).
Consider the outcomes as ordered pairs of the form
(teacher, student)
we could list them as follows
11 12 13 14 15 16
21 22 23 24 25 26
...
71 72 73 74 75 76
81 82 83 84 85 86
there are 48 outcomes
15 have teacher < student
6 have teacher = student
27 have teacher > student
P(teacher wins) = 27/48 = 9/16
A. P(A) = 4/8 = 1/2 (since there are 4 out of 8 possible outcomes where the teacher wins)
B. P(A u B) = P(A) + P(B) - P(A n B)
To find P(B), we need to know the number of sides on the six-sided die that have a 3. Assuming all sides are equally likely, this would be 1/6.
P(B) = 1/6
P(A n B) = P(B) = 1/6
P(A u B) = P(A) + P(B) - P(A n B) = 1/2 + 1/6 - 1/6 = 1/2 + 1/6 = 4/6 = 2/3
a. To find the probability of event A, which is "your teacher wins", we need to determine the total number of possible outcomes and the number of outcomes where your teacher wins.
Total outcomes: Since your teacher rolls an eight-sided die and you roll a six-sided die, the total number of possible outcomes is (8 * 6) = 48.
Outcomes where your teacher wins: Your teacher can get any number from 1 to 8, and you can get any number from 1 to 6. So, your teacher wins if he/she rolls any number greater than yours. The number of outcomes where your teacher wins is (8 - 3) = 5.
Therefore, the probability of event A, P(A), can be calculated as:
P(A) = Number of outcomes where your teacher wins / Total number of outcomes
= 5 / 48
b. Now, let's find the probability of event A union B, which is "your teacher wins or you get a 3 on your first roll".
P(A u B) can be calculated by finding the number of outcomes where either event A or event B occurs, divided by the total number of outcomes.
Outcomes where your teacher wins or you get a 3:
- If your teacher wins, there are 5 possible outcomes.
- If you get a 3 on your first roll, there is only 1 possible outcome.
The number of outcomes where either event A or event B occurs is 5 + 1 = 6.
P(A u B) = Number of outcomes where either event A or event B occurs / Total number of outcomes
= 6 / 48
Hence, P(A u B) = 1 / 8.
To answer the questions, we need to understand a few concepts related to probability.
A fair die is one in which each side has an equal probability of landing face-up when rolled. In this case, the teacher rolls an eight-sided die (with numbers 1 to 8) and you roll a six-sided die (with numbers 1 to 6).
a. To find the probability of the event "your teacher wins" (A), we need to calculate the chances of the teacher rolling a higher number than you. There are six possible outcomes for the teacher to win: 7, 8, 6, 7, 8, 8. Since it is a fair die, each outcome has a probability of 1/8. So, the probability of your teacher winning is (6/8) or 3/4.
b. To find the probability of the event "you get a 3 on your first roll" (B), we need to calculate the chances of you rolling a 3. Since you are rolling a fair six-sided die, the probability of rolling a 3 is 1/6.
To find the probability of the union of events A and B (A u B), we need to calculate the chances of either event A or B happening.
Since events A and B are mutually exclusive (they cannot occur at the same time), to find P(A u B), we can simply add the individual probabilities of A and B.
P(A u B) = P(A) + P(B) = 3/4 + 1/6 = 9/12 + 2/12 = 11/12.
Therefore, the probability of either your teacher winning (event A) or you getting a 3 on your first roll (event B) is 11/12.