Probability Psy 3010 Winter 2017 Extra Credit #6 Due: 2/27 or 2/28 1. If a singl

Question

Probability Psy 3010 Winter 2017 Extra Credit #6 Due: 2/27 or 2/28 1. If a single card is drawn at random from a shuffled deck, what is the probability that it will be a) a jack or a queen b) an ace or a red card 2. If a die is tossed three times, what is the probability of getting a) one six b) anything but a six on every toss c) at least one six 3. If three cards are drawn with replacement from a shuffled deck, what is the probability that a) all three will be aces b) none of the three will be aces c) at least one of them will be an ace 4. Calculate the probabilities in #3 ifeach card is not replaced after it is drawn. 5. A reception was held for local men and women who own either a large business or a small business. There are 200 people in the room: 50 of the people own a large business, 80 of the people are women, and 90 ofthe people are men who own a small business. a) Prepare a joint frequency table of the data. Make sure the numbers in each cell and the totals in the margins are whole numbers. Label the table appropriately. b) Prepare a joint probability table for the data. Be sure to include marginal probabilities. c) Suppose that you randomly select one person in the room, what is the probability that the person is a male? d) Suppose that you randomly select one person in the room. what is the probability that the person is a female small business owner? e) Suppose that you randomly select one person in the room, given that the person owns a large business, what is the probability that the person is a man? 6. If p(A) .50, p(B) .30, and p(A and B) .15: a) Are A and B mutually exclusive events? Why or why not? b) Are A and Bindependent events? Why or why not?

Explanation / Answer

Solution :-

1). If a single card is drawn at random from a shuffled deck,

a). P(a jack or a queen) =

Total number of jack is 4 out of 52 cards.

Total number of queen is 4 out of 52 cards

Number of favourable outcomes i.e. ‘a king or a queen’ is 4 + 4 = 8 out of 52 cards.

Therefore, probability of getting ‘a jack or a queen’

   Number of favorable outcomes
P(E) =   Total number of possible outcome

= 8/52
= 2/13

b). P(an ace or a red card) =

Let event A: the card is an Ace.

Let event B: the card is Red.

So, the probability that the card is either an ace or a red card = P (A U B) = P (A) + P (B) - P (A and B)

= 4/52 + 26/52 - 2/52

= 28/52 = 7/13

The answer is 7/13 = 0.5385

2). If a die is tossed three times,

a). P(one six) =

Note a fair dice means there is one in six chance of hitting any number.

Therefore to hit a six the probability is 1/6

The probability of not hitting a six is 5/6

Since we are doing 3 tosses, it could be 6** or *6* or **6

where * represents any number except a six

hence the probability of getting only a six in three tosses is 3 × ( 1/6 × 5/6 × 5/6 ) = 3 x (25/216) = 0.34722

b). P(atleast one 6) =

The easier approach would be to calculate the chance of not rolling a 6 - that's just 5/6 for the first die, and 5/6 for the second die, and similarily 5/6 for the third die, so by the product rule (as the events are independent), the probability is 5/65/6.5/6 = 125/216.

Then the probability of rolling a 6 is 1 minus the probability of not rolling a 6, which we just calculated: so it is 1125/216 = (216 - 125) / 216 = 0.421.

3). If the cards are drawn with replacement then every time at the denominator we get 52 as the number of of cards in deck is 52.

For eg - P(ace) - 4/52

P(all three ace) = 3 * 4/52 = 0.23

P(non is ace) = 1 - P(at least one ace) = 1 - (4/52 * 48/52 * 48/52 * 48/52)

= 1 - 442368 / 7311616

= 0.061

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