JustasS

A casual laborer has a 70% chance of finding work on each day that she reports to the day labor marketplace. What is the probability that she will work three days out of five?

A) 0.6045.

B) 0.3192.

C) 0.3087.

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P(3) = 5! / [(5 – 3)! × 3!] × (0.73) × (0.32) = 0.3087 = 5 →2nd→ nCr → 3 ×  0.343  × 0.09

JustasS

Assume 30% of the CFA candidates have a degree in economics. A random sample of three CFA candidates is selected. What is the probability that none of them has a degree in economics?

A) 0.343.

B) 0.027.

C) 0.900.

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3! / (0!3!) (0.3)0 (0.7)3 = 0.343

cross-ied

The Night Raiders, an expansion team in the National Indoor Football League, is having a challenging first season with a current win loss record of 0 and 4. However, the team recently signed four new defensive players and one of the team sponsors (who also happens to hold a CFA charter) calculates the probability of the team winning a game at 0.40. Assuming that whether the team wins a game is independent of whether it wins any other game, the probability that the team will win 6 out of the next 10 games is closest to:

A) 0.417.

B) 0.112.

C) 0.350.

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Use the formula for a binomial random variable to calculate the answer to this question. We will define "success" as the team winning a game. The formula is:

p(x) = P(X = x) = [number of ways to choose x from n] × px × (1 - p)n-x,

where [number of ways to choose x from n] = n! / [(n - x)! × x!].
Here, p(x) = P(X = 6) = [10! / (10 − 6)! × 6!] × 0.406 × (1 − 0.40)10-6
= 210.0 × 0.00410 × 0.12960 = 0.11159, or approximately 0.112.
To calculate factorial using your financial calculator: On the TI, factorial is [2nd] ¡→ [x!]. On the HP, factorial is [g] → [n!]. To compute 10! on the TI, enter [10] → [2nd] → [x!] = 3,628,800. On the HP, use [10] → [ENTER] → [g] → [n!].

cross-ied

The number of days a particular stock increases in a given five-day period is uniformly distributed between zero and five inclusive. In a given five-day trading week, what is the probability that the stock will increase exactly three days?

A) 0.333.

B) 0.167.

C) 0.600.

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If the possible outcomes are X0,1,2,3,4,5), then the probability of each of the six outcomes is 1 / 6 = 0.167.

cross-ied

Which of the following random variables would be most likely to follow a discrete uniform distribution?

A) The outcome of a roll of a standard, six-sided die where X equals the number facing up on the die.

B) The number of heads on the flip of two coins.

C) The outcome of the roll of two standard, six-sided dice where X is the sum of the numbers facing up.

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The discrete uniform distribution is characterized by an equal probability for each outcome. A single die roll is an often-used example of a uniform distribution. In combining two random variables, such as coin flip or die roll outcomes, the sum will not be uniformly distributed.

cross-ied

A random variable X is continuous and bounded between zero and five, X0 ≤ X ≤ 5). The cumulative distribution function (cdf) for X is F(x) = x / 5. Calculate P(2 ≤ X ≤ 4).

A) 1.00.

B) 0.40.

C) 0.50.

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For a continuous distribution, P(a ≤ X ≤b) = F(b) − F(a). Here, F(4) = 0.8 and F(2) = 0.4. Note also that this is a uniform distribution over 0 ≤ x ≤ 5 so Prob(2 < x < 4) = (4 − 2) / 5 = 40%.

cross-ied

Which of the following qualifies as a cumulative distribution function?

A) F(1) = 0, F(2) = 0.25, F(3) = 0.50, F(4) = 1.

B) F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0.

C) F(1) = 0.5, F(2) = 0.25, F(3) = 0.25.

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Because a cumulative probability function defines the probability that a random variable takes a value equal to or less than a given number, for successively larger numbers, the cumulative probability values must stay the same or increase.

cross-ied

A cumulative distribution function for a random variable X is given as follows:

x	F(x)
5	0.14
10	0.25
15	0.86
20	1.00

The probability of an outcome less than or equal to 10 is:

A) 14%.

B) 25%.

C) 39%.

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A cumulative distribution function (cdf) gives the probability of an outcome for a random variable less than or equal to a specific value. For the random variable X, the cdf for the outcome 10 is 0.25, which means there is a 25% probability that X will take a value less than or equal to 10.

cross-ied

In a continuous probability density function, the probability that any single value of a random variable occurs is equal to what?

A) One.

B) Zero.

C) 1/N.

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Since there are infinite potential outcomes in a continuous pdf, the probability of any single value of a random variable occurring is 1/infinity = 0.

cross-ied

If a smooth curve is to represent a probability density function, what two requirements must be satisfied? The area under the curve must be:

A) one and the curve must not fall below the horizontal axis.

B) one and the curve must not rise above the horizontal axis.

C) zero and the curve must not fall below the horizontal axis.

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If a smooth curve is to represent a probability density function, the total area under the curve must be one (probability of all outcomes equals 1) and the curve must not fall below the horizontal axis (no outcome can have a negative chance of occurring).