Characteristics of Axiomatic Approach

Q. In a race between Achilles and tortoise, people assigned probability to Achilles winning and tortoise winning. These probability pairs are listed below. How many of these pairs satisfy the axiomatic approach, assuming only two results are tortoise wins and Achilles wins.

1. $(\frac{1}{2},\frac{1}{2})$
2. $(-\frac{1}{2},1)$
3. $(-\frac{1}{2},\frac{3}{2})$
4. $(1,0)$

Q.

Out of a pack of 52 cards one is lost, from the remainder of the pack , two cards are drawn and are found to be spades .Find the chance that the missing card is a spade ?

1. 11/49
2. 10/49
3. 10/50
4. 11/50

Q. If A and B are two events such that P(A) = $\frac{1}{2}$ and P(B) = $\frac{2}{3}$, then

1. 
2. 
3. 
4. All of the above

Q.

Forrest got a box of chocolate from Jenny. It had n different chocolates. Forrest asked Jenny how many chocolates the box has. Jenny, who is an aspiring data scientist replied, the probability of you getting a KitKat is $\frac{1}{19}$. How many chocolates are there in the box, if each chocolate brand is equally - likely and there is only one chocolate of each brand?

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Q.

Which of the following are the conditions to be satisfied in the axiomatic approach to probability?

1. For any event E, P(E) $\ge$ 0

2. P(S) = 1

3. For two mutually exclusive events E and F, $P(E\cup F)=P\left(E\right)+P\left(F\right)$

4. For two exhaustive events E and F, $P(E\cup F)=P\left(E\right)+P\left(F\right)$

Q. A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are p, q and $\frac{1}{2}$, respectively. If the probability that the student is successful, is $\frac{1}{2}$, then

1. $p=q=1$
2. $p=q=\frac{1}{2}$
3. $p=1,q=\frac{1}{2}$
4. $p=1,q=0$

Q.

A computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces 20% and plant $T_2$ produces 80% of the total computers produced. 7% of computers produced in the factory turn out to be defective. It is known that P(computer turns out to be defective, given that it is produced in plant $T_1)$ = 10P (computer turns out to be defective, given that it is produced in plant $T_2$), where P(E) denotes the probability of an event E. A computer produced in the factory is randomly selected and it does not turn out to be defective. Then, the probability that it is produced in plant $T_2$, is ?

1. $\frac{36}{73}$

2. $\frac{47}{79}$

3. $\frac{78}{93}$

4. $\frac{75}{83}$

Q. If X and Y are two events such that $P\left(X|Y\right)=\frac{1}{2},P\left(Y|X\right)=\frac{1}{3}andP(X\cap Y)=\frac{1}{6}.$ Then, which of the following is/are correct ?

1. $P(X\cup Y)=\frac{2}{3}$
2. X and Y are independent
3. X and Y are not independent
4. $P(X^C\cap Y)=\frac{1}{3}$

Q.

A ship is fitted with three engines $E_1,E_{2}and{E}_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}and\frac{1}{4}$. For the ship to be operational, at least two of its engines must function. Let X denote the event that the ship is operational and let $X_1,X_{2}and{X}_3$ denote, respectively the events that the engines $E_1,E_{2}and{E}_3$ are functioning.
Which of the following is/are true?

1. $P\left(X_1^C|X\right)=\frac{3}{16}$

2. P (Exactly two engines of the ship are functioning | X) $=\frac{7}{8}$

3. $P\left(X|X_2\right)=\frac{5}{16}$

4. $P\left(X|X_1\right)=\frac{7}{16}$

Q. A die is rolled. The probability of getting a positive integer less than 7, is