A dice was rolled a number of times. The outcomes are as shown in the table Number 1 2 3 4 5 6 Outcomes 32 m 25 40 28 45 If the probability of obtaining 2 i...

Answer Details

(a) To find the value of m, we can use the fact that the sum of all the outcomes must equal the total number of times the die was rolled. We know that the sum of all the outcomes is:

32 + m + 25 + 40 + 28 + 45 = 220 + m

We also know that the probability of rolling a 2 is 0.15, which means that the number of times a 2 was rolled is:

0.15 x total number of rolls = 0.15N

Since the only other unknown outcome is m, we can set up an equation:

0.15N + m + 32 + 25 + 40 + 28 + 45 = N

Simplifying this equation, we get:

m = N - 210

(b) To find the number of times the die was rolled, we can use the fact that the sum of all the outcomes must equal the total number of times the die was rolled. From part (a), we know that:

32 + m + 25 + 40 + 28 + 45 = 220 + m

Substituting m with N - 210, we get:

32 + N - 210 + 25 + 40 + 28 + 45 = 220 + N - 210

Simplifying this equation, we get:

N = 210

Therefore, the die was rolled 210 times.

(c) To find the probability of obtaining an even number, we need to find the total number of times an even number was rolled and divide it by the total number of rolls. The even numbers are 2, 4, and 6. From the table, we can see that the number of times a 2 was rolled is m, the number of times a 4 was rolled is 40, and the number of times a 6 was rolled is 45. Therefore, the total number of times an even number was rolled is:

m + 40 + 45 = m + 85

Dividing this by the total number of rolls (which we found to be 210 in part (b)), we get:

(m + 85)/210

We don't know the exact value of m, but we know from part (a) that m = N - 210, so we can substitute:

[(N - 210) + 85]/210

Simplifying this equation, we get:

(N - 125)/420

Substituting N with 210, we get:

(210 - 125)/420 = 85/420 = 17/84

Therefore, the probability of obtaining an even number is 17/84.