Question

When a fair die is rolled n times, the probability of getting at most two sixes is 0.532 correct to three significant figures. (a) Find the value of n. ( Can help without using a GDC or write down steps on how to find answer from GDC. not just stating .. I know the answer is 15 but l need working steps on how to get 15 clear?)

Answer 1

Answer:)

Suppose we roll the dice N times. The probability of getting at most two sixes, is given by:

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)$

Where X is the random variable denoting the number of sixes that we obtain in N throws of the fair die. Then, we see that :

$P(X = 0) = \dfrac{5}{6} \cdot \dfrac{5}{6} \cdots \text{ N times} = \left(\dfrac{5}{6}\right) ^N$

Since, we have 5 choices for not getting six (namely 1, 2, 3, 4 and 5) in each throw, and we throw the die N times, and also each throw is independent of each other, so the probabilities multiply.

Similarly, we see that :

$P(X = 1) = {N \choose 1} \dfrac{1}{6} \cdot \dfrac{5}{6} \cdots \text{(N - 1) times} = \dfrac{N}{6} \cdot \left(\dfrac{5}{6}\right) ^{N-1}$

since we get one six and rest all throws give non-sixes, but there are N possible throws this one 6 could occur at.

And also:

$P(X = 2) = {N \choose 2} \left(\dfrac{1}{6}\right)^2 \cdot \dfrac{5}{6} \cdots \text{(N - 2) times} = \dfrac{N(N-1)}{72} \cdot \left(\dfrac{5}{6}\right) ^{N-2}$

since we get two sixes and rest all throws give non-sixes, but there are ${N \choose 2}$ possible throws these two sixes could occur at.

Thus, we are given that :

$\left(\dfrac{5}{6}\right) ^N + \dfrac{N}{6} \cdot \left(\dfrac{5}{6}\right) ^{N-1} + \dfrac{N(N-1)}{72} \cdot \left(\dfrac{5}{6}\right) ^{N-2} \approx 0.532$

and we need to solve for N.

We can at this point use a graphing calculator to graph the function on the left, and see where it intersects the line y = 0.532, as done below:

10 AV p = 0.532 SyS step 5 (15.005, 0.532) -10 -5 15
The red line is the equation on the left, and the blue line is the equation y = 0.532. We see that the intersection occurs at $x \approx 15.005 \Rightarrow N = 15$