Multiplication Rule We will work with the example of picking poker cards out of

Question

Multiplication Rule

We will work with the example of picking poker cards out of a deck. A poker deck contains four suits: diamonds, hearts, spades, and clubs. The diamonds and hearts are red and the spades and clubs are black. Each suit has thirteen cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. This makes a total of 52 cards. A face card will be defined to be a Jack, Queen, or King.

Please put all probabilities as fractions in lowest terms for these problems

Multiplication Rule for Independent events:

1. Select two cards from a deck by selecting one, replacing it in the deck, and then selecting another. What is the probability that the first card is red?

2. Select two cards from a deck by selecting one, replacing it in the deck, and then selecting another. What is the probability that the second card is a Queen?

3. Select two cards from a deck by selecting one, replacing it in the deck, and then selecting another. What is the probability that a red card is selected and then a Queen? Notice the two events “first red” and “second Queen” are independent since the occurrence of one does not affect the other (because the first card is replaced before selecting the second).

Multiplication Rule for non-independent events:

1. Select a card from the deck, and without replacing it, select another. What is the probability both are red? (Think about the total number of possibilities and the number of successes left in the deck

At Least One:

1. A manufacturer of cell phones finds that approximately 2% of their cell phones are defective. If you have 5 friends with this cell phone what is the probability at least 1 of the cell phones will eventually be defective? Please use decimal form

Explanation / Answer

1) P(First card is red) = 26/52 = 1/2

2) Since these are independent events (with replacement) P(second card is Queen) = 4/52 = 1/13

3) P(first red, then queen) = (1/2) x (1/13) = 1/26

1) P( 1st and 2nd red) without replacement.

P(first card is red) = 26/52 = 1/2

P(second card is red) = 25/51

Therefore the required probability = (1/2) x (25/51) = 25/102

1) P(At least 1). = P(1) + P(2) + P(3) + P(4) + P(5).

P(defective cell) = 0.02, P(Non defective) = 1 - 0.02 = 0.98

P(1) = Choose 1 friend out of 5 in 5C1 = 5 ways. The probability his cell is defective = (0.02)1 and the probability that the other 4 are non defective = (0.98)4.

P(1) = 5C1 x (0.02)1 x (0.98)4 = 0.0922368

P(2) = 5C2 x (0.02)2 x (0.98)3 = 0.0037648

P(3) = 5C3 x (0.02)3 x (0.98)2 = 0.0000768

P(4) = 5C4 x (0.02)4 x (0.98)1 = 0.0000008

P(5) = 5C5 x (0.02)5 x (0.98)0 = 0.0

Therefore P(1) + P(2) + P(3) + P(4) + P(5) = 0.0961

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