Module -1

Basic Probability

Question and Answer

1. In poker, a full house (3 cards of one rank and two of another, e.g. 3 fours and 2 queens) beats a flush (five cards of the same suit).

 Solution- A player is more likely to be dealt a flush than a full house. We will be able to precisely quantify the meaning of “more likely” here.

2.     A coin is tossed repeatedly.

Solution- Each toss has two possible outcomes:

Heads (H) or Tails (T)

Bothequally likely. The outcome of each toss is unpredictable; so is the sequence of H and T.

3.     A factory has two machines A and B making 60% and 40% respectively of the total production. Machine A produces 3% defective items, and B produces 5% defective items. Find the probability that a given defective part came from A.

Solution- We consider the following events:

A: Selected item comes from A.

B: Selected item comes from B.

D: Selected item is defective.

We are looking for . We know:

Now,

So we need

Since, D is the union of the mutually exclusive events and (the entire sample space is the union of the mutually exclusive events A and B)

4.     Two fair dice are rolled, 1 red and 1 blue. The Sample Space is

S = {(1, 1),(1, 2), . . . ,(1, 6), . . . ,(6, 6)}.Total -36 outcomes, all equally likely (here (2, 3) denotes the outcome where the red die show 2 and the blue one shows 3).

(a) Consider the following events:

A: Red die shows 6.

B: Blue die shows 6.

Find , and .

Solution:

NOTE:so for this example. This is not surprising - we expect A to occur in of cases. In of these cases i.e. in of all cases, we expect B to also occur.

5.     You are given a bag of marble. Inside the bag are 5 red marble, 4 white marble, 3 blue marble. Calculate the probability that with 6 trials you choose 3 marbles that are red, 1 marble that is white and 2 marble is blue. Replacing each marble after it is chosen.

Solution:

6.     You are randomly drawing cards from an ordinary deck of cards. Every time you pick one you place it back in the deck. You do this 5 times. What is the probability of drawing 1 heart, 1 spade, 1club, and 2 diamonds?

Solution:

7.     A plant places biscuits into boxes of 100. The probability that a biscuit is cracked is 0.03. Find the probability that a box contains 2 cracked biscuits

Solution

This is a binomial distribution by means of n = 100 besides p = 0.03.

These standards are external the range of the counters and include lengthy calculations.

Using the Poisson estimate (test: np = 100 x 0.3 = 3, which is less than 5)

Let X be the random variable of the number of cracked biscuits

The mean λ = np = 100 × 0.3 = 3

P(X = 2) = 0.224 (from counters)

8.     If the probability that a light bulb is defective is 0.8, what is the probability that the light bulb is not defective?

Solution:

Probability that the bulb is defective, p = 0.8

Probability that the bulb is not defective, q = 1 - p = 1 - 0.8 = 0.2

9.     The effect of the degree of risk on the portfolio is illustrated as follows: Portfolio risk is measured in this example when the correlation of the coefficients is -1, 1.5, -0.5, 9, + 1 when x = - 1, the risk is the lowest, the risk would be zero if the percentage of inversion in values   X1 and Xj is modified so that the standard deviation becomes 0 and x = -1.

(i)When

(ii)When

(iii)When

(iv)When,

10.  

Let  be a random variable such that

Find a lower bound to its variance.

Solution

The lower bound can be derived thanks to Chebyshev's inequality: