Statistics
Suppose that two cards are randomly selected from a standard 52-card deck. (a) What is the probability that the first card is a spade and the second card is a spade if the sampling is done without replacement? (b) What is the probability that the first card is a spade and the second card is a spade if the sampling is done with replacement?
1) without replacement:- the probability that first card is a club = 13C1/52C1 = 1/4 the probability that second card is also a club = 12C1/51C1 = 12/51 = 4/17 thus, the overall probability is ,P = (1/4)*(4/17) = 1/17 2) with replacement:- the probability that first card is a club = 13C1/52C1 = 1/4 the probability that second card is also a club = 13C1/52C1 = (1/4) thus, the overall probability is ,P = (1/4)*(1/4) = 1/16
Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E={1, 2, 4}.
3/10 = 0.3
A golf ball is selected at random from a golf bag. If the golf bag contains 7 type A balls, 8 type B balls, and 5 type C balls, find the probability that the golf ball is not a type A ball.
7+8+5 = 20 /( Type B + Type/Total Balls)
Suppose that E and F are two events and that P(E and F)=0.2 and P(E)=0.8. What is P(F|E)?
=0.2/0.8 =0.25
What is the probability of obtaining seven tails in a row when flipping a coin? Interpret this probability.
0.00781
Suppose there is a 19.7% probability that a randomly selected person aged 35 years or older is a smoker. In addition, there is a 11.1% probability that a randomly selected person aged 35 years or older is female, given that he or she smokes. What is the probability that a randomly selected person aged 35 years or older is female and smokes? Would it be unusual to randomly select a person aged 35 years or older who is female and smokes?
0.197 x 0.111 = 0.022 Yes, because it is less than 0.5%
Find the probability of the indicated event if P(E)=0.30 and P(F)=0.55. Find P(E and F) if P(E or F)= 0.75 Solve the general addition rule equation for P(E and F). The solved equation is shown below. P(E and F)=P(E)+P(F)−P(E or F)
0.30+0.55-0.75 = 0.10
What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero on empirical rules. Does this mean that the event is impossible?
0; No
Determine whether the events E and F are independent or dependent. Justify your answer. (a) E: A person attaining a position as a professor. F: The same person attaining a PhD.
E and F are dependent because attaining a PhD can affect the probability of a person attaining a position as a professor. E cannot affect F and vice versa because the people were randomly selected, so the events are independent.
Find the probability P(Ec) if P(E) = 0.39
For any event that is to occur its non-occurrence is the complimentary event. The sum of probability of any event and its compliment is one. 1 - 0.39 = 0.61
Two events E and F are ________ if the occurrence of event E in a probability experiment does not affect the probability of event F.
Independent
You suspect a 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the following table. Do you think the die is loaded? Why?
No, because each value has an approximately equal chance of occuring.
Is the following a probability model? What do we call the outcome "blue"? Color/Probability Red/0.25 Green/0.1 Blue/0 Brown/0.3 Yellow/0.3 Orange/0.1 What do we call the probability of blue?
No, because the probabilities do not sum to 1; impossible event
If E and F are not disjoint events, then P(E or F)=________.
P(E)+P(F) - P(E and F)
Which of the following numbers could be the probability of an event? 0.03, 1.55, 0.39, -0.4, 0, 1
Recall that the probability of any event E, P(E), must be greater than or equal to 0 and less than or equal to 1. 0.03,0.39,0,1
The data represent the number of driving fatalities for a certain area by age for male and female drivers. a) What is the probability that a randomly selected driver fatality who was female was greater than 70 years old? b) What is the probability that a randomly selected driver fatality who was greater than 70 was female? c)
a) add all of the female numbers and divide by 1731 (female drivers) =15990/1731 =0.108 b)Add male and female 70 year old drivers divide by # of female 70 year olds 1731/4897 = 0.353 c) divide each greater than 70 (m & f) by total male: 3166/4897 = 0.647 female: 1731/4897 = 0.353
For the fiscal year 2007, a tax authority audited 1.65% of individual tax returns with income of $100,000 or more. Suppose this percentage stays the same for the current tax year. What is the probability that two randomly selected returns with income of $100,000 or more will be audited?
p = 1.65% 1.65/100 =0.0165 P=p^2 =(0.0165)^2 =0.000272
If a person rolls a six-sided die and then flips a coin, describe the sample space of possible outcomes using 1, 2, 3, 4, 5, 6 for the die outcomes and H, T for the coin outcomes. (Make sure your answers reflect the order stated.)
{1H,2H,3H,4H,5H,6H,1T,2T,3T,4T,5T,6T}
A probability experiment is conducted in which the sample space of the experiment is S={4,5,6,7,8,9,10,11,12,13,14,15}. Let event E={5,6,7,8,9,10} and event F={9,10,11,12}. List the outcomes in E and F. Are E and F mutually exclusive?
{9,10}; No, E and F have outcomes in common
A probability experiment is conducted in which the sample space of the experiment is S={5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, event E={6, 7, 8, 9, 10} and event G={11, 12, 13, 14}. Assume that each outcome is equally likely. List the outcomes in E and G. Are E and G mutually exclusive?
{}, Yes, because the event E and G have no outcomes in common.