Answer to Question #198621 in Statistics and Probability for Tashmi

Question

6. Two ordinary six-sided dice were tossed. Set up a sample space for this experiment and

hence find the probability that

i. Sum of the points on the two dice is 7,

ii. Points on the first die are greater than the points on the second die,

iii. First die shows an even number,

iv. Points on the dice are the same, (i.e. throwing a double),

v. Difference in the outcomes of two dice is 2.

7. Suppose that two machines I and II in a factory operate independently of each other. Past

experience showed that during a given 8-hour time, machine I remains inoperative one

third of the time and machine II does so about one fourth of the time. What is the

probability that at least one of the machines will become inoperative during the given

period?

8. Two coins are tossed. If A is the event “head on the first coin”, B is the event “head on

the second coin” and C is the event “coins fall alike”, show that the events A, B, and C

are pair wise independent but not completely independent.

Answer 1

6.

Sample space for two dice (outcomes):

i)

$P(sum=7)=6/36=1/6$

ii)

$P(first \ is\ greater)=15/36=5/12$

iii)

$P(first\ is\ even)=18/36=1/2$

iv)

$P(first \ and\ second\ are\ same)=6/36=1/6$

v)

$P(difference\ is\ two)=8/36=2/9$

7.

Probability that machine I remains inoperative:

$P_1=1/3$

Probability that machine II remains inoperative:

$P_2=1/4$

The probability that at least one of the machines will become inoperative during the given

period:

$P=P_1+P_2+P_1\cdot P_2=1/3+1/4+1/12=8/12=2/3$

8.

Probabilities:

$P(A)=P(B)=P(C)=1/2$

$P(A\land B)=P(A)\cdot P(B)$

$P(B\land C)=P(B)\cdot P(C)$

$P(A\land C)=P(A)\cdot P(C)$

So, the events A,B,C are pair wise independent.

But

$P(A\land B\land C)\ne P(A)\cdot P()\cdot P(C)$

So, the events A,B,C are not completely independent.