# Some definitions and properties of Probability

- Definitions
**Mutually Exclusive:**If one event occurs, then other cannot occur**Exhaustive:**All exhaustive events taken together form the complete sample space (Sum of probability = 1)**Independent Events:**One event occurring has no effect on the other event

- The probability of any event A:

- If the probability of happening of event A is P(A), then the probability of A not happening is (1-P(A))
- For example, if the probability of a company going bankrupt within one year period is 20%, then the probability of company surviving within next one year period is 80%

For a bond with “B” rating, assume 1 year probability of default for each issuer is 6%, and that default probability of each issuer are independent. What is the probability that both issuers avoid default during the 1st year.

88.0%

88.4%

94.0%

96.4%

**B.**

Both would avoid default only if None defaults

This implies that first does not default AND second does not default

= (1 – PD (first)) x (1 – PD (second))

= (1 – 0.06) x (1 – 0.06) = 0.884 = 88.4%

# Some Properties of Probability

The probability of happening of event A or event B can be given as the sum of the three portions defined by the figure below:

Jensen, a portfolio manager is managing two portfolios. One for High Net Worth Individuals (HNI) and second for Low Net Worth Individuals (LNI)

HNI portfolio contains 5 bonds and 7 stocks and LNI contains 6 bonds and 11 stocks

One instrument from HNI is transferred to LNI portfolio

Now Jensen selects an instrument from LNI, what is the probability that instrument selected is stock?

- 0.5382
- 0.7821
- 0.6435
- None of these

**C.**

Here required probability = [P(stock transferred from HNI) AND P(Stock selected from LNI)] OR

[P(bond transferred from HNI) AND P(Stock selected from LNI)]

So, the required probability = (7/12) × (12/18) + (5/12) × (11/18) = 139/216 = 0.6435

Hence option ‘C’ is correct

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