Probability Questions (please help!!)

Question

1) EXPECTED VALUE. The state sells 5 million lottery tickets at $2.00 a piece.

100 = 6000 winners 2,500=250𝑤𝑖𝑛𝑛𝑒𝑟𝑠

2,500=250winners

50,000 = 12 winners 100,000=6𝑤𝑖𝑛𝑛𝑒𝑟𝑠

100,000=6winners

250,000 = 3 winners

$2,500,000 = 1 winner

You buy one ticket. What is the probability that your ticket wins a prize? (any prize)

What is the expected value of your ticket?

2) STANDARD DEVIATION. Two towns in the mountains of Greece dispute which one of them has the most geniuses. They agree to settle the argument by having the International Genius Testing Committee administer an IQ test, where a score of 150 or higher means the test-taker is a genius. The town of Aristotle selects its 100 smartest citizens to take the test, and they score a mean of 125 with a standard deviation of 25. The town of Pythagoras selects its 600 smartest citizens to take the test, and they score a mean of 130 with a standard deviation of 10.

Based on what you know about statistics, say which town has the most geniuses and support your answer.

3) PROBABILITY. A fair coin is tossed (heads or tails) 6 times. What is the probability that the coin will land “heads” exactly 3 out of the 6 times?

What is the probability that the exact sequence of the 6 tosses will be HEADS-TAILS-HEADS-TAILS-HEADS-TAILS ?

What is the probability that the exact sequence of the 6 tosses will be TAILS-TAILS-TAILS- HEADS-HEADS-HEADS ?

4) COUNTING. How many unique ways can the letters of the word LUMBERJACKS be arranged?

How many unique ways can the letters of the word HIGHLIGHT be arranged?

How many unique ways can the letters COOKBOOK be rearranged?

What is 88! divided by 86! ?

Answer 1

1) Expected Value -

Probability that you win a prize = 0.001

Expected value to ticket = 0.12 + 0.125 + 0.12 + 0.12 + 0.15 + 0.5 = $1.135

2) We need to find out how many people have a IQ of more than 150 in each town.

For first town, Z score for 150 is = (150-125)/25 i.e. 1.

Now p value for this is 1-0.8413 i.e. 0.1587.

Thus, number of geniuses are 100*0.1587 i.e. 16.

For second town, Z score for 150 is = (150-130)/10 = 2.

So p value = 1-0.9772 = 0.0228.

Thus number of geniuses = 600*0.0228 = 14.

So town of Aristotle has more number of geniuses.

3)Probability will be 6C3*(0.5^3)(0.5^3) = 0.3125.

Probability will be 0.5^6 = 0.015

Probability will be 0.5^6 = 0.015.

4) Number of ways is 11! = 39916800.

Number of ways is 9!/(3!*2!*2!) = 15120.

Number of unique ways = 8!/(4!*2!) = 840.

88!/86! = 88*87 = 7656.