What are examples of variables that follow a binomial probab

What are examples of variables that follow a binomial probability distribution? What are examples of variables that follow a Poisson distribution? When might you use a geometric probability?

A. Discrete Probability Variables

If we flip a fair die 1,000 times, what would we expect the mean number of dots to be?

If we flipped a coin 10 times, what is the probability that we would get five heads?

Using the variables you choose describe a potential real-life situation that would use that variable and probability.

B. Discrete Probability and a Deck of Cards

From a single deck of cards, you select a card, look at it, put it back in the deck, and shuffle. If you do this 10 times, what is the probability of drawing exactly (1) one ace, (b) two aces, (c) three aces, (d) four aces (each of these is calculated separately)? Out of the 10 draws of cards, what is the mean number of aces that you would expect to find?

What would be the probability of finding three or four aces out of the 10 draws? How is this calculation different than the ones above?

Try to do a similar problem with face cards.

When might this type of issue come up in real life? Certainly, we don\'t deal with cards often!

Solution

A)

The expected value of each flip is given by, as each side has probability of 1/6,

E(x) = 1/6 (1) + 1/6 (2) +1/6 (3) +1/6 (4) +1/6 (5) +1/6 (6) = 3.5

Thus, for 1000 flips, the epected number of dots is

1000 E(x) = 1000(3.5) = 3500 [ANSWER]

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A.2

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.5      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.24609375

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REAL LIFE SITUATION: For example, a stop light is green 30% of the time. What is the probability that in 10 times you pass in that intersection, that 7 times it is green?

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