AP Statistics Binomial and Geometric Probability

A friend has placed a large number of plastic disks in a hat and invited you to select one at random.  He informs you that half are red and half are blue.  If you draw a disk, record the color, replace it, and repeat 100 times, which of the following is true?

• It is unlikely you will choose red more than 50 times.

• If you draw 10 blue disks in a row, it is more likely you will draw a red on the next try.

• The overall proportion of red disks drawn should be close to 0.50.

• The chance that the 100th draw will be red depends on the results of the first 99 draws.

A claimed psychic was presented with 200 cards face down and asked to determine if the card was one of five symbols: a star, cross, circle, square, or three wavy lines.  The "psychic" was correct in 50 cases.  To determine if he has ESP, we test the hypothesis H₀:  p = 0.20, Hₐ: p > 0.20, where p represents the true proportion of cards for which the psychic would correctly identify the symbol in the long run.  Assume the conditions for inference are met.  The P-value is 

• between .10 and .05

• between .05 and .025

• between .025 and .01

• between .01 and .001

Joe reads that 1 our of 4 eggs contains salmonella bacteria. So he never uses more than 3 eggs in cooking. If eggs do or don't contain salmonella independently of each other, the number of contaminated eggs when Joe uses 3 chosen at random has the following distribution:

• Binomial; n = 4 and p = 1/4

• Binomial; n = 3 and p = 1/4

• Binomial; n = 3 and p = 1/3

• Geometric; p = 1/4

• Geometric; p = 1/3

A binomial event has n = 60 trials. The probability of success on each trial is 0.4. Let X be the count of successes of the event during the 60 trials. What are mean X and standard deviation X?

• 24; 3.79

• 24; 14.4

• 4.9; 3.79

• 4.9; 14.4

The probability that an American household will have at most 3 cars is...

• 0.93

• 0.20

• 0.13

• 0.07

The probability that an American household will have at least 2 cars is...

• 0.55

• 0.35

• 0.45

• 0.93

What's the expected number of cars in a randomly selected American household?

• 1.00

• 1.75

• 1.84

• 2.00

How would you interpret the expected value in number 3?

• The maximum number of cars for the household is Exp(C).

• The average number of cars for the household is Exp(C).

• If we randomly select many households, the average number of cars per household will be approximately by Exp(C).

• If we randomly select many households, the average number of cars per household will vary approximately by Exp(C).

A deck of cards contains 52 cards , of which 4 are aces. You are offered the following wager: Draw one card at random from the deck. You win $10 if the card drawn is an ace. Otherwise, you lose $1. If you make this wager very many times, what will be the mean amount you win?

• About -$1, because you will lose most of the time.

• About $9, because you win $10 but lose only $1.

• About -$0.15; that is, on average you lose about 15 cents.

• About $0.77; that is, on average you win about 77 cents.

• About $0, because the random draw gives you a fair bet.

A certain vending machine offers 20-ounce bottles of soda for $1.50. The number of bottles X bought from the machine on any day is a random variable with mean 50 and standard deviation 15. Let the random variable Y equal the total revenue from this machine on a given day. Assume the machine works properly and that no sodas are stolen from the machine. What are the mean and standard deviation of Y?

• Mean Y = $1.50 and St. Dev Y = $22.50

• Mean Y = $1.50 and St. Dev Y = $33.75

• Mean Y = $75 and St. Dev Y = $18.37

• Mean Y = $75 and St. Dev Y = $22.50

• Mean Y = $75 and St. Dev Y = $33.75