Two number cubes are rolled- one blue and one yellow. Find the probability that the blue cube is less than 3 and the product is 8.
a) Is this event independent or dependent? Explain why.
b) Find the probability that the blue cube is less than 3 and the product is 8.
a) Well, this event is actually dependent. Why, you ask? Because the outcome of the first roll of the blue cube affects the possibilities for the second roll. So, it's like the blue cube is saying to the yellow cube, "Hey, buddy, here's what number I rolled, now it's your turn to do something with it!"
b) Now let's get to the calculation part. To find the probability that the blue cube is less than 3 and the product is 8, we need to consider both rolls. There are 6 possible outcomes for each cube, so that gives us a total of 36 possible outcomes.
For the blue cube to be less than 3, it can either be a 1 or a 2. There are 2 out of 6 possibilities for that, which is a probability of 2/6 or 1/3.
Now, for the product to be 8, we need the yellow cube to be a 4. There is only 1 out of 6 possibilities for that, which is a probability of 1/6.
Since these two events are dependent, we multiply the probabilities together: (1/3) * (1/6) = 1/18.
So, the probability that the blue cube is less than 3 and the product is 8 is 1/18. And remember, no matter how small that probability is, it's always greater than the probability of finding a unicorn in your backyard!
a) This event is dependent. The outcome of the blue cube being less than 3 affects the outcome of the product being 8. If the blue cube is not less than 3, then it is not possible for the product to be 8.
b) To find the probability that the blue cube is less than 3 and the product is 8, we need to find the number of favorable outcomes and divide it by the total number of possible outcomes.
The blue cube can be less than 3 in two ways: it can be 1 or 2. The yellow cube has to be 8 divided by the value of the blue cube, so if the blue cube is 1, the yellow cube has to be 8, and if the blue cube is 2, the yellow cube has to be 4.
Therefore, there is only one favorable outcome: (1, 8) or (2, 4).
The total number of possible outcomes is the number of outcomes for each cube multiplied together. Since each cube has 6 possible outcomes, the total number of outcomes is 6 x 6 = 36.
So the probability is 1/36.
a) To determine if this event is independent or dependent, we need to consider whether the outcome of one roll affects the outcome of the other roll. In this case, we have two independent events: the outcome of the blue cube roll and the outcome of the yellow cube roll. The result of the blue cube roll does not impact the yellow cube roll or vice versa. Therefore, this event is independent.
b) To find the probability that the blue cube is less than 3 and the product is 8, we need to determine the favorable outcomes and the total outcomes.
Favorable outcomes:
The blue cube can be less than 3 in two ways: it can be 1 or 2. The yellow cube can be 8 only when it is rolled as 4. So, the favorable outcome is the combination of these possibilities: blue cube = 1, yellow cube = 8 and blue cube = 2, yellow cube = 4.
Total outcomes:
Both the blue cube and the yellow cube have 6 possible outcomes since they are regular number cubes.
Therefore, the probability of the blue cube being less than 3 and the product being 8 is:
Number of favorable outcomes / Total number of outcomes
= 2 (number of favorable outcomes) / 36 (total number of outcomes)
= 1 / 18
So, the probability is 1/18.
(Problem a) The events are dependent because you're looking for a specific product.
(Problem b) Now, assuming that the cubes have six faces:
P(blue is less than 3)= 2/6 = 1/3
The product is 8; 1*8=8, but there is no 8 on a die. So blue must be 2 and yellow must be 4; 2*4=8. Since there is a 1/6 chance for each die (Six faces per die), you multiply 1/6*1/6 = 1/36 chance.
Now add the P(blue is less than 3) = 1/3
1/36 + 1/3 = ?
1/36 + 12/36 = 13/36
Therefore, P(blue is less than 3 and the product is 8)= 13/36
Hope this helps :)