Statistics and Probability Test

Result

Statistics and Probability Test - 4

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
2 / -0.83

Let A and B be any two arbitrary events, then, which one of the following is true?

A
P(A ∩B) = P(A) P(B)

B
P(A ∪B) = P(A) + P(B)

C
P(A | B) = P(A ∩B)/P(B)

D
P(A ∪ B ) < P(A) + P(B)

Solution
Question 2
2 / -0.83

A bag contains 10 white balls and 15 black balls. Two balls are drawn in succession. The probability that one of them is black and the other is white is

A
2/3

B
4/5

C
1/2

D
1/3
Solution
To find the probability that one ball is black and the other is white when drawing two balls in succession, we can use the following steps:
The total number of balls is 25 (10 white + 15 black).
The possible outcomes for drawing one black and one white ball are:
First ball white, second ball black.
First ball black, second ball white.
Calculating the probabilities:
Probability of drawing a white ball first and a black ball second:
P(White first) = 10/25
P(Black second | White first) = 15/24
Combined probability = (10/25) ×(15/24) = 150/600
Probability of drawing a black ball first and a white ball second:
P(Black first) = 15/25
P(White second | Black first) = 10/24
Combined probability = (15/25) ×(10/24) = 150/600
Now, add the probabilities of both scenarios:
Total probability = 150/600 + 150/600 = 300/600 = 1/2
Thus, the probability that one ball is black and the other is white is 1/2 .
Question 3
2 / -0.83

Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is

A
1/36

B
1/3

C
25/36

D
11/36

Solution

P(atleast one of dice will have 6 facing
= 1 - P(none of dice have 6 facing up)
Question 4
2 / -0.83

The probability that top and bottom cards of a randomly shuffled deck are both aces is-

A

B

C

D

Solution

The probability that the bottom card of a randomly shuffled deck is ace = 4/52 because there are 4 aces out of total 52 cards.
From the remaining 3 aces out of SVcards the probability that the top card is also an ace = 3/51.
So required probability
Question 5
2 / -0.83

E₁ and E₂ are events in a probability space satisfying the following constraints:

The value of Pr(E₁ ), the probability of the event E₁ , is

A
0

B
1/4

C
1/2

D
1

Solution

Constraints are
Question 6
2 / -0.83

A program consists of two modules executed sequentially. Let f₁ (t) and f₂ (t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by

A

B

C

D

Solution

Let the time taken for first and second modules be represented by x and y and total time = t.
∴ t = x + y is a random variable.
Now the joint density function

which is also called as convolution of f₁ and f₂ , abbreviated as f₁ * f₂ .
Question 7
2 / -0.83

A sample space has two events A and B such that probabilities

What is P(A ∪B)?

A
11/12

B
10/12

C
9/12

D
8/12

Solution

We know that:
Question 8
2 / -0.83

If the difference between the expectation of the square of a random variable (E[x² ] and the square of the expectation of the random variable (E[x])² is denoted by R, then

A
R = 0

B
R <0

C
R ≥0

D
R >0
Solution
Random variable assigns a real number to each possible outcome.
Let X be a discreet random variable,then

where V(x) is the variance of x,
Explanation:
The difference between the expectation of the square of a random variable (E[X2]) and the square of the expectation of the random variable (E[X])2 is called the variance of a random variable
Variance measure how far a set of numbers is spread out
A variance of zero(R=0) indicates that all the values are identical
A variance of X = R =E[X2]- (E[X])² This quantity is always non-negative as it is an expectation of a non-negative quantity
A non-zero variance is always positive means R >0
So, R ≥0 is the answer. Since variance is and hence never negative,
Question 9
2 / -0.83

Consider a random variable Xthat takes values +1 and -1 with probability 0.5 each. The values of the cumulative distribution function F(x) at x = -1 and +1 are

A
0 and 0.5

B
0 and 1

C
0.5 and 1

D
0.25 and 0.75

Solution

The p.d.t of the random variable is

The cumulative distribution function F(x) is the probability uptox as given below:
Question 10
2 / -0.83

If P and Q are two random events, then the following is TRUE

A
Independence of P and Q implies that probability (P ∩Q) - 0

B
Probability (P ∪Q) ≥Probability (P) + Probability (Q)

C
If P and Q are mutually exclusive, then they must be independent

D
Probability ( P ∩Q) ≤Probability (P)

Solution

(a) is false since of Pand Q are independent

which need not be zero.
(b) is false since

(c) is false since independence and mutually exclusion are unrelated properties.
(d) is true