# The tutor offers points about combining probabilities.

My gut reaction, thinking about when to add probabilities, is that it’s done less often than multiplying. However, there is one obvious type of situation in which you add:

**Example 1**

Each ticket bought has 1/1000 probability of winning a new PC. Josh buys five tickets. What is his probability of winning?

Solution:

In this case, Josh’s probability of winning is the sum of each of his tickets’ winning chances:

P(Josh wins)=1/1000+1/1000+1/1000+1/1000+1/1000=5/1000=1/200

Note that if one ticket wins, the others can’t. Such a premise – where if event X happens, then event Y cannot – is called *mutual exclusivity*. In turn, we refer to X and Y as *mutually exclusive* events. Normally, when probabilities are added, they belong to mutually exclusive events, either of which will achieve the same outcome.

Generally, probabilities are multiplied under the following conditions:

- The events happen in sequence, or else they
*both* happen.
- The events are independent of one another; ie, if one event happens, it doesn’t affect the other’s likelihood of occurring.

**Example 2**

At a fishing hole, trout is caught with 55% probability; whitefish, 32%. Assuming you catch two fish, what is the probability that the first is a trout, then the second, a whitefish?

Solution:

We assume there are so many fish present that catching one does not affect the probability of which kind you’ll catch next.

P(TW)=P(T)xP(W)=(0.55)(0.32)=0.176

Next we look at a problem whose solution needs both operations:

**Example 3**

A husband and wife have a joint chequing account. Each cheque needs only one signature; either can sign. The cheques are all written in either black or blue. The wife prefers blue ink; 80% of the time she signs in blue. The husband prefers black; 75% of the time, he signs in black. The wife is more interested in managing the account: in total, 85% of the cheques are written by her.

What is the probability that a cheque from the couple is signed in black?

Solution:

Since only one person signs the cheque, we can add the probabilities of either doing so:

P(Black)=P(wife wrote it, signed in black)+P(husband wrote it, signed in black)

Recall that the wife writes 85% of the cheques, and signs in black only 20% of the time:

P(wife wrote the cheque and signed it in black)=0.85×0.20

If the wife writes 85% of the cheques, the husband must write the other 15% of them. He signs in black 75% of the time:

P(husband wrote the cheque and signed it in black)=0.15×0.75

Therefore,

P(Black)=P(wife wrote it, signed in black)+P(husband wrote it, signed in black)

becomes

P(Black)=0.85×0.20 + 0.15×0.75=0.17 + 0.1125=0.2825

Apparently, the probability a given cheque will be in black is 0.2825, or 28.25%.

I’ll be talking much more about probability in coming posts:)

Source:

Ross, Sheldon. __A First Course in Probability__. New York: Macmillan, 1988.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

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