Determine which of the following represent valid probability distributions
$\displaystyle{\begin{array}{c|c|c|c|c}x & 0 & 1 & 2 & 3 \\\hline P(x) & 1/8 & 3/8 & 3/8 & 1/8\end{array}}$
$\displaystyle{\begin{array}{c|c|c|c|c}x & 0 & 1 & 2 & 3 \\\hline P(x) & 1/8 & -3/8 & 3/4 & 1/2\end{array}}$
$\displaystyle{\begin{array}{c|c|c|c|c}x & 0 & 1 & 2 & 3 \\\hline P(x) & 1/8 & 3/8 & 3/8 & 2/8\end{array}}$
Is the function described below a pdf? Explain.
$$f(x)=\frac{x+1}{10} \textrm{ for } x = 0, 1, 2, 3$$Find the mean, variance, and standard deviation for the following distribution
$$\displaystyle{\begin{array}{c|c|c|c|c}x & 0 & 1 & 2 & 3 \\\hline P(x) & 1/8 & 3/8 & 3/8 & 1/8\end{array}}$$A game at a local fair will give $\$$100 to anyone that can break a balloon by throwing a dart at it. The game costs $\$$5 to throw a single dart, and you're willing to spend $\$$30 trying to win. Assuming that you have a 10% chance of hitting the balloon on any given throw, find the expected number of darts you will throw.
Consider the possible outcomes, we could throw a dart 1, 2, 3, 4, 5, or 6 times. Letting these be the possible values for $x$, we find $P(x)$, the probability that we throw exactly $x$ darts for each such value.
We throw exactly 1 dart precisely when we break the balloon on our first attempt. Thus, $P(1) = 0.10$. We throw exactly 2 darts when we fail to break the balloon on the first attempt, but succeed on the second. So $P(2) = (0.90)(0.10)$, recalling that if we have a 10% chance of hitting the balloon on any given throw, we have a 90% chance of missing it. Similarly, we find $P(3) = (0.90)^2(0.10)$, $P(4) = (0.90)^3(0.10)$, and $P(5) = (0.90)^4(0.10)$.
However, we find the probability of throwing 6 darts a little differently, as one throws exactly 6 darts when one misses the previous 5. It doesn't matter whether we hit the balloon on the sixth try or not, as we are only willing to spend $\$30$ trying to win, and will stop throwing darts after the sixth try regardless. Consequently $P(6) = (0.90)^5$.
Collecting these results in a table, we have:
$$\begin{array}{c|c|c|c|c|c|c} x & 1 & 2 & 3 & 4 & 5 & 6\\\hline P(x) & 0.10 & (0.90)(0.10) & (0.90)^2(0.10) & (0.90)^3(0.10) & (0.90)^4(0.10) & (0.90)^5 \end{array}$$Now, calculating the expected value $\mu$, we have
$$\begin{array}{rcl} \mu &=& (1)(0.10) + (2)(0.90)(0.10) + (3)(0.90)^2(0.10)\\ & & + \, (4)(0.90)^3(0.10) + (5)(0.90)^4(0.10) + (6)(0.90)^5\\ &=& 4.68559 \end{array}$$Based on previous information, it has been determined that 90% of the population brush their teeth once a day. Answer the following for a sample of 20 people:
Treat as a binomial distribution, with $n = 20, p = 0.90,$ and $q = 0.10$:
(a) $$P(18) = {}_{20}C_{18} (0.90)^{18}(0.10)^2$$ (b) $$\begin{array}{rcl} P(18) + P(19) + P(20) &=& {}_{20}C_{18} (0.90)^{18}(0.10)^2\\ && + \, {}_{20}C_{19} (0.90)^{19}(0.10)^1\\ && + \, {}_{20}C_{20} (0.90)^{20}(0.10)^0 &\approx& 0.6769 \end{array}$$A fair coin is tossed eight times. Let the random variable be the number of heads that appear.
Toss a coin 16 times. Let X be the number of heads that appear.
Roll a standard die 8 times. Let $X$ be the number of 2's rolled. Find the following:
A shipment of 25 computers contains 10 computers with a defective DVD burner. What is the probability, if a random sample of 6 computers is selected and then tested, that the sample will contain at least 1 defective computer?
In a club of 25 members there are 10 married men and 15 unmarried men. What is the probability that a subcommittee of 6 will have at least 1 married man?
It is known that 5% of all tax returns contain at least one error. For a random selection of 10 tax returns, what is the probability that at most 2 of them contain errors?
Assume 75% of Americans wear seatbelts. If 200 Americans are selected at random, find the expected number of people in this group that wear their seatbelts.
Toss a fair coin 100 times. Let X be the number of heads showing. Give the mean and the standard deviation for this experiment.
The probabilities that a game of chance results in a win, loss, or tie for the player to go first is 0.48, 0.46, and 0.06, respectively. If the game is played 8 times, find the probability that there will be 3 wins, 4 losses and 1 tie.
A DVD has a defect on average every 2 inches along its track. What is the probability of seeing less than 3 defects within a 5 inch section of its track?
Usually 50 potential jurors are held to compose a jury of 12. Suppose that this group of 50 has 15 females and 35 males.
Compare and contrast the Poisson distribution with the Binomial distribution.
The probability that a worker will become disabled in a one-year period is 0.0045. If there are 500 workers on an assembly line, find the probability that more than 4 workers will become disabled. (Use the Poisson distribution to approximate the probability.)
Assume that customers enter a large store at the rate of 60 per hour (one per minute). Find the probability that during a five-minute interval no one will enter the store.
Given the function defined below,
$$f(x) = \frac{x+1}{6} \textrm{ for } x=0,1,2$$You have a box with 25 different colored balls as follows: 4 red, 6 blue, 10 white, and 5 green.
Hypergeometric.
$\displaystyle{\textrm{(a) } \frac{({}_{4}C_1)({}_{6}C_1)({}_{10}C_1)({}_{5}C_1)}{{}_{25}C_4} = \frac{1200}{12650} \approx 0.095}$
$\displaystyle{\textrm{(b) } \frac{({}_{10}C_2)({}_{15}C_0)}{{}_{25}C_2} = \frac{45}{300} = 0.15}$
Assume that the probability of a college student having a car on campus is 0.30. A random sample of 12 students is taken. What is the probability that at least 4 will have a car on campus?
A bridge hand contains 13 cards. What is the probability that a bridge hand will contain 9 spades, all four aces, and one non-spade, non-ace.
The switchboard receives an average of 3 calls per minute. For a randomly selected minute, what is the probability that there will be at least 4 calls?