Question

1. Prove that a3 mod c = (a mod c)3 mod c where a and c are positive integers.

Show the steps clearly and provide your logic where necessary.




2. Suppose you have two cars. One car has 6 engines and the other has 4 engines. Each
engine of the car fails with a probability of 0.25. The engines are independent of each
other. That means their success or failure does not depend on others. You know that your
each car continues to operate if at least 50% of its’ engines operate. You decided to go on
a long drive but you are not sure which car you should take. So you decided to determine
the probability of success of each car. Based on your calculation of the probability which
car you should take?



3. A hardworking ant moves 30 centimeter per minute. It starts at one end of a rubber band of
length one meter. After each minute, a little kid stretches the rubber band by one meter. During
the stretching operation, the ant maintains its’ relative position. Is it possible for the ant to reach
the finish of the rubber band? If it is possible, then after which minute the ant will surpass the
end?

Explain your logic behind the calculation.




4. Suppose, you have been analyzing Neymar’s performance in each football match. You found that
Neymar can score a maximum of 3 goals in a single match. He can score less but not more than

three. You also found that if Neymar does not score any goal in a match, then he cannot score
more than one goal in the next match. In such a case the probability of scoring one goal is 0.4. If
he scores one goal in a match, then he scores 0, 1, 2 or 3 goals with the probabilities of 0.2, 0.6,
0.1, and 0.1, respectively. If he scores two goals in a match, then he scores 0, 1, 2 or 3 goals with
probabilities of 0.1, 0.5, 0.2, and 0.3, respectively. If he scores a hat-trick (i.e., 3 goals) in a
match, then he does not remain goalless in the next match. In such a case, his probabilities of scoring other number of goals are equal. Now, in a tournament, you see that Neymar scores two

goals in the first match. What will be the probability that Neymar will score one goal in the third

match?​

Answer 1

Answer:

Explanation:

I dont know