# Joint probability/Conditional Probability

The oasis outpost of Abu Ilan, in the heart of the Negev desert, has a population of 20 Bedouin tribesmen and 20 Farima tribesmen. El Kamin, a nearby oasis, has a population of 32 Bedouins and 8 Farima. A lost Israeli soldier, accidentally separated from his army unit, is wandering through the desert and arrives at the edge of one of the oases. The soldier has no idea which oasis he has found, but the first person he spots at a distance is a Bedouin. What is the probability that he wandered into Abu Ilahn? What is the probability that he is in El Kamin?

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#### Solution Preview

Bedouin Farima Total

Abu Ilan 20 20 40

El Kamin 32 8 40

Probability of Spotting a bedouin = Probability of selecting Abu Ilan x Probability of selecting a Bedouin in Abu Ilan +Probability of selecting El Kamin x Probability of selecting a Bedouin in El ...

#### Solution Summary

The solution calculates conditional probability: Probability that the oasis is Abu Ilan given that the person spotted is a Bedouin & Probability that the oasis is El Kamin given that the person spotted is a Bedouin.

Normal Probability based on Z score

A soft drink bottling company maintains records concerning the number of unacceptable bottles of soft drink obtained from the filling and capping machines. Based on past data, the probability that a bottle came from machine I and was nonconforming is 0.05 and the probability that a bottle came from machine II and was nonconforming is 0.035. These probabilities represent the probability of one bottle out of the total sample having the specified characteristics. Half the bottles are filled on machine I and the other half are filled on machine II.

a. Give an example of a simple event.

b. Give an example of a joint event.

c. If a filled bottle of soft drink is selected at random, what is the probability that it is a nonconforming bottle?

d. If a filled bottle of soft drink is selected at random, what is the probability that it was filled on machine II?

e. If a filled bottle of soft drink is selected at random, what is the probability that it was filled on machine I and is a conforming bottle?

3. According to Investment Digest ("Diversification and the Risk/Reward Relationship", Winter 1994, 1-3), the mean of the annual return for common stocks from 1926 to 1992 was 14.4%, and the standard deviation of the annual return was 20.5%. During the same 67-year time span, the mean of the annual return for long-term government bonds was 5.5%, and the standard deviation was 7.0%. The article claims that the distributions of annual returns for both common stocks and long-term government bonds are bell-shaped and approximately symmetric. Assume that these distributions are distributed as normal random variables with the means and standard deviations given previously.

a. Find the probability that the return for common stocks will be greater than 0%.

b. Find the probability that the return for common stocks will be less than 0%.

c. Find the probability that the return for common stocks will be less than 15%.

d. Find the probability that the return for common stocks will be greater than 20%.

e. Find the probability that the return for common stocks will be greater than 30%.

f. Find the probability that the return for common stocks will be less than -10%.

Hint: There are many ways to attack this third problem in the HW. If you would like the normal distribution table so you can draw the pictures

See attached file for full problem description.

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