3. (a) A fair die is tossed eight times. Suppose A is the event of getting at le

Question

3. (a) A fair die is tossed eight times. Suppose A is the event of getting at least three 5's on the eight tosses of the die, while B is the event of getting precisely three 5's on the eight tosses of the die. i. Determine with reason if the events A and B are mutually exclusive. ii. Determine the probabilities of the events A and B. Are the events A and B independent? (b) Suppose a fair coin is tossed 7 times. Let X be the discrete random variable representing the number of heads which can occur. i. Determine the discrete probability distribution ofX ii. Determine the Expected value ? = E(X) of X. ii. Determine the Variance, Var(x), of X iv. Determine the Standard Deviation, ?, of X

Explanation / Answer

Solution:

a.

i. The events A and B are mutually exclusive because both cannot occur together.

ii. P (A) = The probability of atleast three 5's is

P (X ? 3) = 1 - P (X < 3)

= 1 - [P (X = 0) + P (X = 1) + P (X = 2)]

= 1 - [8C0 (1/6)^0 (5/6)^8 + 8C1 (1/6)^1 (5/6)^7 + 8C2 (1/6)^2 (5/6)^6]

= 1 - 0.8652

= 0.1348

P (B) = the probability of three 5's

P (X = 3) = 8C3 (1/6)^3 (5/6)^5

P (X = 3) = 0.1042

Hence, they are dependent.

b. Using binomial distribution with p (head) = 1/2 = 0.5 and n = 7

P (X = x) = 7Cx (1/2)^x (1/2)^7-x, x = 0, 1 , 2 ............7

i.

ii. E (X) = ?X.P(X)

E (X) = 3.5

iii. Var (X) = ?X^2.P(X) - [?X.P(X)]^2

Var (X) = 14 - (3.5)^2

Var (X) = 1.75

iv. SD (X) = Sqrt(Variance)

SD (X) = Sqrt(1.75)

SD (X) = 1.323

X P(X) X.P(X) X^2.P(X) 0 0.007813 0 0 1 0.054688 0.054688 0.054688 2 0.164063 0.328125 0.65625 3 0.273438 0.820313 2.460938 4 0.273438 1.09375 4.375 5 0.164063 0.820313 4.101563 6 0.054688 0.328125 1.96875 7 0.007813 0.054688 0.382813 Sum = 3.5 14

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