there is a six sided die, number 1- to 6. what is the probability that it it take exactly two tries to roll a 4? what is the expected waiting time until and odd prime number is rolled?
what is the probability of not getting a four on the first try and a 4 on the second.
Pr=5/6 * 1/6
odd prime number? 1,3, 5, ?
expected waiting time? Don't know, what is the time for each roll?
Prob one roll=3/6
since this is 1/2, then one should "expect" a prime number in the first roll.
To calculate the probability that it takes exactly two tries to roll a 4 on a six-sided die, we need to consider the possible outcomes.
In the first roll, we have a 1/6 chance of rolling a 4. If this happens, we are done and it took only one try.
However, if we don't roll a 4 in the first try, we need to roll again. On the second roll, we also have a 1/6 chance of rolling a 4.
Since these two events are independent, we can multiply the probabilities together. Therefore, the probability of rolling a 4 on the second try is (1/6) * (1/6) = 1/36.
However, we also need to account for the possibility that we roll something other than a 4 on the first try and then never roll a 4 in successive tries. This means we need to calculate the probability of not rolling a 4 on the first try and also not rolling a 4 on the second try.
The probability of not rolling a 4 on any single try is 5/6 (since there are five other numbers on the die). Since we want to calculate the probability of not rolling a 4 on either the first or the second try, we multiply this probability by itself: (5/6) * (5/6) = 25/36.
Finally, we combine the probabilities of rolling a 4 on the first try (1/6), rolling a 4 on the second try (1/36), and not rolling a 4 on either try (25/36).
Therefore, the probability that it takes exactly two tries to roll a 4 on a six-sided die is (1/6) + (1/36) + (25/36) = 12/36 = 1/3.
Now let's move on to the expected waiting time until an odd prime number is rolled.
Odd prime numbers on a six-sided die are 3 and 5. We need to consider the probabilities of rolling each number and the expected number of rolls it takes to roll each specific number.
For rolling a 3: The probability of rolling a 3 on any single roll is 1/6. The expected number of rolls until rolling a 3 is 1/(1/6) = 6 rolls.
Similarly, for rolling a 5: The probability of rolling a 5 on any single roll is also 1/6. The expected number of rolls until rolling a 5 is also 1/(1/6) = 6 rolls.
Now we calculate the average of these expected values. Since the probabilities of rolling a 3 and 5 are the same, we simply take the average of 6 rolls and 6 rolls, which is (6 + 6) / 2 = 12 / 2 = 6 rolls.
Therefore, the expected waiting time until an odd prime number is rolled on a six-sided die is 6 rolls.