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.