You roll two five-sided dice. The sides of each die are numbered from 1 to 5. The dice are “fair" (all sides are equally likely), and the two die rolls are independent.
Part (a): Event A is “the total is 10" (i.e., the sum of the results of the two die rolls is 10).
Is event A independent of the event “at least one of the dice resulted in a 5"?
- unanswered
Is event A independent of the event “at least one of the dice resulted in a 1"?
- unanswered
Part (b): Event B is “the total is 8."
Is event B independent of getting “doubles" (i.e., both dice resulting in the same number)?
- unanswered
Given that the total was 8, what is the probability that at least one of the dice resulted in a 3?
part (a)
1 and 2 No.
part (b)
1. No.
2. Looks like conditional probability (i failed this question. This answer was not used back in the day)
scenarios:
(the only possibility for an 8)
3,5 =0.2*0.2
4,4=0.2*0.2
5,3=0.2*0.2
at least one of the dices was a 3?
So thus = 0.08/0.12 =0.66666667
hope it helps!
Well, it seems like you have a lot of unanswered questions regarding probability and events with dice. Let's see if I can inject some humor while answering them!
Part (a): Is event A independent of the event "at least one of the dice resulted in a 5"?
Hmm, let me think, I'm rolling the dice in my head... Well, if you have a 5 on one of the dice, it certainly narrows down the possibilities for getting a total of 10, doesn't it? So, I'm afraid event A is not independent of the event "at least one of the dice resulted in a 5." They are like a couple that depends on each other to go to the "10th" level of love!
Is event A independent of the event "at least one of the dice resulted in a 1"?
Oh boy, it's like the dice rolled off the table and faceplanted into a 1, isn't it? Well, same as before, having a 1 on one of the dice limits the options for getting a total of 10. So, once again, event A is not independent of the event "at least one of the dice resulted in a 1." They stick together like a loyal pair of dice on a gambling night!
Part (b): Is event B independent of getting "doubles"?
Ah, doubles, the perfect match! Like two peas in a pod. Well, if you manage to get doubles, it doesn't really tell us anything about the total sum being 8, does it? So, event B is independent of getting doubles. It's like having a friend who's always there for you, but won't help you win any rolls!
Given that the total was 8, what is the probability that at least one of the dice resulted in a 3?
Well, isn't it funny how things work out sometimes? Given that the total is 8, it means we could have a 3 on one die and a 5 on the other, or vice versa! So, to calculate the probability, we need to count all the combinations that feature at least one 3 when the total is 8. And remember, don't worry too much about the numbers; in the end, it's all just a roll of the dice!
To determine whether events A and B are independent, we need to check if the probability of event A occurring is affected by the occurrence of event B, and vice versa.
For Part (a):
1. To calculate the probability of event A, we need to determine the number of ways to get a sum of 10 on two five-sided dice. Let's list the possible outcomes:
(4, 6), (5, 5), (6, 4)
So, there are 3 ways to get a sum of 10.
2. The probability of event A occurring can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Since each die has 5 possible outcomes, and the dice rolls are independent, the total number of possible outcomes is 5 * 5 = 25. Therefore, the probability of event A is 3/25.
3. Now, let's analyze whether event A is independent of the event "at least one of the dice resulted in a 5." To do this, we need to find the probability of event A occurring given that at least one of the dice resulted in a 5.
If at least one of the dice resulted in a 5, the possible outcomes are:
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5)
From these outcomes, we can see that there is still only one way to get a sum of 10: (5, 5).
Therefore, the probability of event A occurring given that at least one of the dice resulted in a 5 is 1/12.
4. Since the probability of event A given that at least one of the dice resulted in a 5 is not equal to the probability of event A, event A is dependent on the occurrence of "at least one of the dice resulted in a 5."
5. Note that we could repeat the same steps to determine the independence of event A and the event "at least one of the dice resulted in a 1."
For Part (b):
1. Since we are only interested in the case where the total is 8, let's find the possible outcomes that result in this sum:
(3, 5), (4, 4), (5, 3)
2. The probability of event B can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. As mentioned earlier, the total number of possible outcomes is 5 * 5 = 25. Therefore, the probability of event B is 3/25.
3. Now, let's analyze whether event B is independent of the event "getting doubles." To do this, we need to find the probability of event B occurring given that we get doubles.
If we get doubles, the possible outcomes are:
(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)
From these outcomes, we can see that there is only one way to get a sum of 8: (4, 4).
Therefore, the probability of event B occurring given that we get doubles is 1/5.
4. Since the probability of event B given that we get doubles is equal to the probability of event B, event B is independent of the occurrence of "getting doubles."
Given that the total was 8, we can determine the probability that at least one of the dice resulted in a 3 by considering all the possible outcomes resulting in a sum of 8, which are (3, 5), (4, 4), and (5, 3). Out of these three outcomes, two of them have at least one dice resulting in a 3, namely (3, 5) and (5, 3).
Therefore, the probability that at least one of the dice resulted in a 3 given that the total was 8 is 2/3.
To determine whether two events are independent, we need to check if the occurrence of one event affects the probability of the other event.
Let's explore each part of the question:
Part (a) - Event A: The total is 10.
Event A can occur in three ways: (4, 6), (5, 5), or (6, 4).
Event "at least one of the dice resulted in a 5":
This event can occur in four ways: (5, 1), (5, 2), (5, 3), or (5, 4).
To verify independence, we need to check if the probability of event A given event "at least one of the dice resulted in a 5" is the same as the probability of event A alone.
To calculate the probability of event A, we divide the number of favorable outcomes (2: (4, 6) and (6, 4)) by the total number of possible outcomes (25 = 5 * 5).
Probability of A: P(A) = 2/25
To calculate the probability of event "at least one of the dice resulted in a 5":
There are 4 possible outcomes out of 25 (5 * 5). The outcomes are (5, 1), (5, 2), (5, 3), and (5, 4).
Probability of "at least one of the dice resulted in a 5": P(B) = 4/25
To calculate the probability of event A given event "at least one of the dice resulted in a 5", we need to find the number of outcomes where both conditions are satisfied. In this case, the only outcome is (5, 5).
Probability of A given B: P(A|B) = 1/25
Since P(A) is not equal to P(A|B), event A is not independent of the event "at least one of the dice resulted in a 5".
For the second part of the question, to determine independence with the event "at least one of the dice resulted in a 1", you can follow the same steps and calculations.
Part (b) - Event B: The total is 8.
The ways event B can occur are: (2, 6), (3, 5), (4, 4), (5, 3), or (6, 2).
Event "getting doubles":
The outcomes when getting doubles are: (1, 1), (2, 2), (3, 3), (4, 4), or (5, 5).
To check independence, we need to compare the probabilities of event B and event "getting doubles" alone with the probability of event B given the occurrence of "getting doubles".
Again, calculate the probabilities as explained earlier and check if they are equal for independence.
Given that the total is 8, we want to find the probability that at least one of the dice resulted in a 3.
Out of the five possible outcomes for an 8, only two outcomes (3, 5) and (5, 3) satisfy the condition of at least one dice being a 3.
Probability of at least one dice resulting in a 3: P(3) = 2/5
Therefore, the probability that at least one of the dice resulted in a 3, given that the total is 8, is 2/5.