Answer:
The expected value of rolling a 5 or 6 is 10.
Step-by-step explanation:
The sample space of rolling a standard number cube is:
S = {1, 2, 3, 4, 5 and 6}
The cube is standard, this implies that each side has an equal probability of landing face-up.
So, the probability of all the six outcomes is same, i.e.
.
Now it is provided that Apu rolls the cube <em>n</em> = 30 times.
Let the random variable <em>X</em> represent the value on the face of cube.
The event of rolling a 5 and rolling a 6 are mutually exclusive, i.e. they cannot occur together.
So, P (X = 5 and X = 6) = 0.
Compute the probability of getting a 5 or 6 as follows:
P (X = 5 or X = 6) = P (X = 5) + P (X = 6) - P (X = 5 and X = 6)
= P (X = 5) + P (X = 6)
![=\frac{1}{6}+\frac{1}{6}\\\\=\frac{2}{6}\\\\=\frac{1}{3}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B1%7D%7B6%7D%2B%5Cfrac%7B1%7D%7B6%7D%5C%5C%5C%5C%3D%5Cfrac%7B2%7D%7B6%7D%5C%5C%5C%5C%3D%5Cfrac%7B1%7D%7B3%7D)
Compute the expected value of rolling a 5 or 6 as follows:
![E(X = 5\ \text{or}\ X = 6)=n\times P(X = 5\ \text{or}\ X = 6)](https://tex.z-dn.net/?f=E%28X%20%3D%205%5C%20%5Ctext%7Bor%7D%5C%20X%20%3D%206%29%3Dn%5Ctimes%20P%28X%20%3D%205%5C%20%5Ctext%7Bor%7D%5C%20X%20%3D%206%29)
![=30\times \frac{1}{3}\\\\=10](https://tex.z-dn.net/?f=%3D30%5Ctimes%20%5Cfrac%7B1%7D%7B3%7D%5C%5C%5C%5C%3D10)
Thus, the expected value of rolling a 5 or 6 is 10.