OpenStudy (anonymous):

A die is rolled four times, what is the chance that not all the rolls show three or more spots?

3 years ago
OpenStudy (anonymous):

i think 1\2

3 years ago
OpenStudy (anonymous):

Can someone explain why to me?

3 years ago
OpenStudy (anonymous):

um because 12/24 is 1/2 . you multiply 6 by 4 to get 24 , an since it say not all the rolls are 3 or more you take half of 6 which is 3 and then simplify. But like because you multiply 6 by 4 you gotta do the same to the 3

3 years ago
OpenStudy (whpalmer4):

Let's make sure we are in agreement what the problem is asking. My interpretation of the problem is that we roll a six-sided die (numbered 1-6) 4 times, and we want to know the probability that when all 4 rolls are done, 1 or more of the rolls will have produced a 1 or a 2. Agreed?

3 years ago
OpenStudy (whpalmer4):

We can view this as the complement of the probability that each of the 4 rolls will produce a 3, 4, 5, or 6.

3 years ago
OpenStudy (whpalmer4):

If you roll the die once, what is the probability that you will roll a 3, 4, 5 or 6? What is the probability you will roll a 1 or 2?

3 years ago
OpenStudy (whpalmer4):

Rolling the die repeatedly produces a set of independent events; the outcome of the second one is not affected by the first, the outcome of the third is not affected by the second, etc. To combine the probabilities of two independent events, we multiply the probabilities. For example, if we flip a fair coin, the probability of the coin coming up heads is 1/2, and the probability of it coming up tails is also 1/2. Those are the two possible outcomes, and the sum of all of the probabilities is 1. If we flip a fair coin twice, the probability that we will get heads on both flips (two independent events) is 1/2 * 1/2 = 1/4. Similarly, the probability that we will get tails on both flips is also 1/2 * 1/2 = 1/4. The remaining possible outcomes are 1 heads and 1 tails and 1 tails and 1 heads. Notice that we have 4 possible outcomes. The probability of any given combination of outcomes is simply the fraction of the number of such outcomes divided by the total number of outcomes. Our possible coin flip results: heads heads heads tails tails heads tails tails We have 4 possible outcomes. One of them is 2 heads, so the probability of that is 1/4, just like we got via multiplication. Another one is two tails, so the probability of that is also 1/4, again just like we got via multiplication. Finally, we have 2 outcomes where we end up with neither all heads nor all tails, so the probability of that happening is 2/4. If we subtract the probabilities of the other two outcomes from 1, we get \[1-\frac{ 1 }{ 4 }-\frac{1}{4} = \frac{2}{4} = \frac{1}{2}\]

3 years ago
OpenStudy (whpalmer4):

Our coin flipping could be viewed as a simplified version of the die rolling problem. Suppose we flip a coin 3 times, and want to find the probability that we will not have it come up heads each time. Probability for each flip of getting heads is 1/2. The probability of not having it come up heads each time is 1-probability of getting heads each time. That's a set of independent events, so we can find the probability of 3 successive flips giving us heads by multiplying the probability of 1 flip giving us heads with itself: \[P_{3 heads} = P_{heads}*P_{heads}*P_{heads} = \frac{1}{2} *\frac{1}{2}*\frac{1}{2} = \frac{1}{8}\] The probability of getting 3 heads in a row (\(P_{3heads})\) is \(\frac{1}{8}\). That means the probability of not getting 3 heads in a row is \[1-P_{3heads} = 1-\frac{1}{8} = \frac{7}{8}\]

3 years ago
OpenStudy (whpalmer4):

I hope this explanation will help you determine the correct answer to your problem. If it doesn't, another way you could approach your problem is to make a table showing all of the possible rolls of the die: 1 1 1 1 1 1 1 2 1 1 1 3 .... 6 6 6 6 and counting how many of the rows do not have all of the rolls showing 3 or more spots, and dividing that by the total number of rows. The answer will be exactly the same.

3 years ago
OpenStudy (anonymous):

Thank you to both of you! The answer I obtained is 65/81. Would this be correct?

3 years ago