Suppose you roll two standard number cubes. Let event a be rolling a four on the first die and let event b be rolling a three on the second die. are a and B independent events? are A and B mutually exclusive events? Explain
they are independent, since the rolls do not affect each other
they are clearly not mutually exclusive, since both can happen
To determine whether events A and B are independent or mutually exclusive, we need to understand the definitions of these terms and compare them to the given events.
1. Independent events: Two events A and B are considered independent if the outcome of one event has no impact on the probability of the other event occurring. To test for independence, we can check if the probability of both events happening together is equal to the product of their individual probabilities.
In this case, event A is rolling a four on the first die. Since a standard number cube has six sides, each with an equal chance of landing, the probability of rolling a four on the first die is 1/6.
Event B is rolling a three on the second die. Again, the probability of this is 1/6 since each side of the die has an equal likelihood.
To find the probability of both A and B occurring together, we multiply their individual probabilities:
P(A and B) = P(A) * P(B)
P(A and B) = (1/6) * (1/6)
P(A and B) = 1/36
As the probability of both events happening together is 1/36, and the product of their individual probabilities is also 1/36, we can conclude that events A and B are independent.
2. Mutually exclusive events: Two events A and B are mutually exclusive if they cannot happen at the same time. In other words, if one event occurs, then the other event cannot occur.
In this case, event A is rolling a four on the first die, and event B is rolling a three on the second die. Since the outcomes of the two dice rolls are independent, it is possible for both event A and event B to happen simultaneously. For example, if you roll a four on the first die and a three on the second die, both events A and B occur. Therefore, events A and B are not mutually exclusive.
In summary:
- Events A and B are independent because the probability of both events occurring together (rolling a four on the first die and a three on the second die) is equal to the product of their individual probabilities.
- Events A and B are not mutually exclusive because they can happen at the same time.
To determine if events A and B are independent, we need to check if the occurrence of one event affects the probability of the other event. Here, event A is rolling a four on the first die, and event B is rolling a three on the second die.
To find out if A and B are independent, we need to determine two things:
1. The probability of event A occurring.
2. The probability of event B occurring.
1. The probability of rolling a four on the first die:
A standard number cube has six sides with numbers 1, 2, 3, 4, 5, and 6. The probability of rolling a four is 1 out of 6, as there is one favorable outcome (rolling a four) out of six equally likely outcomes.
So, P(A) = 1/6.
2. The probability of rolling a three on the second die:
Again, a standard number cube has six sides, so the probability of rolling a three is also 1 out of 6, just like in event A.
P(B) = 1/6.
Now, let's find the probability of both A and B occurring simultaneously (P(A and B)).
Since these events are happening on two different dice, the results are independent. So, the probability of both events A and B occurring is the product of their individual probabilities:
P(A and B) = P(A) * P(B) = (1/6) * (1/6) = 1/36.
If we compare P(A and B) = 1/36 to P(A) * P(B) = (1/6) * (1/6) = 1/36, we can see that they are equal.
Since P(A and B) = P(A) * P(B), we can conclude that events A and B are independent.
Now, let's determine if events A and B are mutually exclusive.
Mutually exclusive events cannot occur at the same time. In this case, event A is rolling a four on the first die, and event B is rolling a three on the second die.
Since these events are happening on different dice, it is possible to roll a four on the first die and a three on the second die simultaneously. Therefore, events A and B are not mutually exclusive.