Tossing two coins. Find probability to getting either no heads or one head
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
1. Tossing two coins. Find probability to getting either no heads or one head
2. Rolling a single die. Find probability to getting an odd number (1, 3, or 5 )
3. Rolling a fair die and getting a 3. Find both the odds for and the odds against
4. The odds on (against) your bet is 2 to 3. If you bet $15 and win, how much will you gain
5. Suppose you reach in and draw two marbles at random from a bag containing 5 white marbles(W), 5 black marbles(B), and 5 red marbles (R)
a) List all possible outcomes of this process( for example, RR, BW, and so on)
b) Make a table or graph showing the probability distribution for the events of 0, 1, and 2 black marbles.
6. Find the and probability of the events.The next five births at a hospital all being boys.
7. Find the either/or probability of the events. Getting a sum of either 2, 3, 4, or 5 on a roll of two dice.
8. Determine the probability. Getting at least one even number in five rolls of single die.
9. At least how many times do you have to roll a fair die to be sure that the probability of rolling at least one 5 is greater than 9 in 10 (0.9)
10. The data in the following table show the outcomes of guilty and not-guilty pleas in 1028 criminal court cases.
|
|
Guilty plea |
Not-guilty plea |
|
Sent to prison |
492 |
74 |
|
Not sent to prison |
668 |
24 |
a. What is the probability that a randomly selected defendant either pled guilty or was sent to prison?
b. What is the probability that a randomly selected defendant either pled not guilty or was not sent to prison?
11. You are given 5 to 1 odds against tossing three heads with three coins, meaning you win $5 if you succeed and you lose $1 if you fail. Find the expected value. Would you expect to win or lose money in 1 game? In 100 games? Explain
12. An insurance policy sells for $300. Based on past data an average of 1 in 100 policyholders will file a $10, 000 claim, an average of 1 in 250 policyholders will file a $25,000 claim, and an average of 1 in 500 policyholders will file a $50,000 claim.Find the expected value per policy sold. If the company sells 10,000 policies, what is the expected profit or loss?
13. Suppose you play a coin toss game in which you win $1 if a head appears and loses $1 if a tail appears. In the first 100 coin tosses, heads come up 46 times and tails come up 54 times.
a) What percentage of times has heads come up in the first 100 tosses? What is your net gain or loss at this point?
b) Suppose you toss the coin 200 more times( a total of 300 tosses), and at that point, heads have come up 47% of the time. Is this increase in the percentage of heads consistent with the law of large numbers? Explain. What is your net gain or loss at this point?
c) How many heads would you need in the next 100 tosses in order to break even after 400 tosses? Is this likely to occur?
d) Suppose that, still behind after 400 tosses, you decide to keep playing because you are “due “ for a winning streak. Explain how this belief would illustrate the gambler’s fallacy.
14. The table shows the number of accidents, fatalities, hours and miles flew for U.S. airlines.
|
Year |
Accidents |
Fatalities |
Hours flown |
Miles flown |
|
2000 |
59 |
98 |
16.7 million |
7.1 billions |
|
2005 |
36 |
28 |
18.7 |
7.8 |
|
2010 |
29 |
0 |
17.2 |
7.3 |
|
2015 |
25 |
0 |
17.4 |
7.6 |
a. Compute the accident rate per million hours flown in 2000 and 2015. By this measure, has travel on U.S. airlines become safer?
b. Compute the accident rate per billion miles flown in 2000 and 2015. By this measure, has travel on U.S. airlines become safer?
15. The number of death due to diabetes and kidney disease for an American during 2015 is respectively 68,500 and 42,100. What is the relative frequency probability of death due to diabetes for an American during 2015? How much greater is the risk of death due to diabetes than the risk of death due to kidney disease?
16. John needed to buy an Air Conditioner. In the catalog, the Air Conditioner costs $165 with a 15% discount. How much will the Air Conditioner cost?
17. Rolling a single die. Find the probability of getting an even number or a prime number.
18. Calculate the yield on a $1000 Treasury bond with a coupon rate of 6% that has a market value of $860.
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