Bad5shah

Thomas Baynes has applied to both Harvard and Yale. Baynes has determined that the probability of getting into Harvard is 25% and the probability of getting into Yale (his father’s alma mater) is 42%. Baynes has also determined that the probability of being accepted at both schools is 2.8%. What is the probability of Baynes being accepted at either Harvard or Yale, but not both?

A) 7.7%.

B) 64.2%.

C) 10.5%.

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Using the addition rule, the probability of being accepted at Harvard or Yale, but not both, is equal to: P(Harvard) + P(Yale) − P(Harvard and Yale) = 0.25 + 0.42 − 0.028 = 0.642 or 64.2%.

Bad5shah

There is a 30% probability of rain this afternoon. There is a 10% probability of having an umbrella if it rains. What is the chance of it raining and having an umbrella?

A) 40%.

B) 33%.

C) 3%.

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P(A) = 0.30. P(B | A) = 0.10. P(AB) = (0.30)(0.10) = 0.03 or 3%.

Bad5shah

If two events are independent, the probability that they both will occur is:

A) 0.50.

B) Cannot be determined from the information given.

C) 0.00.

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If two events are independent, their probability of their joint occurrence is computed as follows:
P(A∩B) = P(A) × P(B). Since we are not given any information on the respective probabilities of A or B, there is not enough information.

Bad5shah

A very large company has twice as many male employees relative to female employees. If a random sample of four employees is selected, what is the probability that all four employees selected are female?

A) 0.3333.

B) 0.0625.

C) 0.0123.

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Since there are twice as many male employees to female employees, P(male) = 2/3 and P(female) = 1/3. Therefore, the probability of 4 “successes” = (0.333)4 = 0.0123.

Bad5shah

There is a 50% chance that the Fed will cut interest rates tomorrow. On any given day, there is a 67% chance the DJIA will increase. On days the Fed cuts interest rates, the probability the DJIA will go up is 90%. What is the probability that tomorrow the Fed will cut interest rates or the DJIA will go up?

A) 0.33.

B) 0.95.

C) 0.72.

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This requires the addition formula. From the information: P(cut interest rates) = 0.50 and P(DJIA increase) = 0.67, P(DJIA increase | cut interest rates) = 0.90. The joint probability is 0.50 × 0.90 = 0.45. Thus P (cut interest rates or DJIA increase) = 0.50 + 0.67 − 0.45 = 0.72.

Bad5shah

Jessica Fassler, options trader, recently wrote two put options on two different underlying stocks (AlphaDog Software and OmegaWolf Publishing), both with a strike price of $11.50. The probabilities that the prices of AlphaDog and OmegaWolf stock will decline below the strike price are 65% and 47%, respectively. The probability that at least one of the put options will fall below the strike price is approximately:

A) 0.31.

B) 1.00.

C) 0.81.

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We calculate the probability that at least one of the options will fall below the strike price using the addition rule for probabilities (A represents AlphaDog, O represents OmegaWolf):
P(A or O) = P(A) + P(O) − P(A and O), where P(A and O) = P(A) × P(O)
P(A or O) = 0.65 + 0.47 − (0.65 × 0.47) = approximately 0.81

Bad5shah

An analyst has a list of 20 bonds of which 14 are callable, and five have warrants attached to them. Two of the callable bonds have warrants attached to them. If a single bond is chosen at random, what is the probability of choosing a callable bond or a bond with a warrant?

A) 0.70.

B) 0.85.

C) 0.55.

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This requires the addition formula, P(callable) + P(warrants) – P(callable and warrants) = P(callable or warrants) = 14/20 + 5/20 – 2/20 = 17/20 = 0.85.

Bad5shah

A very large company has equal amounts of male and female employees. If a random sample of four employees is selected, what is the probability that all four employees selected are female?

A) 0.0256

B) 0.0625.

C) 0.1600

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Each employee has equal chance of being male or female. Hence, probability of 4 “successes” = (0.5)4 = 0.0625

Bad5shah

Which of the following is a joint probability? The probability that a:

A) company merges with another firm next year.

B) stock increases in value after an increase in interest rates has occurred.

C) stock pays a dividend and splits next year.

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A joint probability applies to two events that both must occur.

Bad5shah

If two fair coins are flipped and two fair six-sided dice are rolled, all at the same time, what is the probability of ending up with two heads (on the coins) and two sixes (on the dice)?

A) 0.8333.

B) 0.0069.

C) 0.4167.

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For the four independent events defined here, the probability of the specified outcome is 0.5000 × 0.5000 × 0.1667 × 0.1667 = 0.0069.