Learning Outcomes
Independent and mutually exclusive do not mean the same thing. Independent EventsTwo events are independent if the following are true:
Two events [latex]A[/latex] and [latex]B[/latex] are independent if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are NOT independent, then we say that they are dependent. Sampling may be done with replacement or without replacement.
If it is not known whether [latex]A[/latex] and [latex]B[/latex] are independent or dependent, assume they are dependent until you can show otherwise.
You have a fair, well-shuffled deck of [latex]52[/latex] cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are [latex]13[/latex] cards in each suit consisting of [latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[/latex], [latex]J[/latex] (jack), [latex]Q[/latex] (queen), [latex]K[/latex] (king) of that suit.
You have a fair, well-shuffled deck of [latex]52[/latex] cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are [latex]13[/latex] cards in each suit consisting of [latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[/latex], [latex]J[/latex] (jack), [latex]Q[/latex] (queen), [latex]K[/latex] (king) of that suit. Three cards are picked at random.
You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of [latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[/latex], [latex]J[/latex] (jack), [latex]Q[/latex] (queen), [latex]K[/latex] (king) of that suit. [latex]S[/latex] = spades, [latex]H[/latex] = Hearts, [latex]D[/latex] = Diamonds, [latex]C[/latex] = Clubs.
Which of [latex]1[/latex] or [latex]2[/latex] did you sample with replacement and which did you sample without replacement? This video provides a brief lesson on finding the probability of independent events.
You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of [latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[/latex], [latex]J[/latex] (jack), [latex]Q[/latex] (queen), [latex]K[/latex] (king) of that suit. [latex]S[/latex] = spades, [latex]H[/latex] = Hearts, [latex]D[/latex] = Diamonds, [latex]C[/latex] = Clubs. Suppose that you sample four cards without replacement. Which of the following outcomes are possible? Answer the same question for sampling with replacement.
Mutually Exclusive Events[latex]A[/latex] and [latex]B[/latex] are mutually exclusive events if they cannot occur at the same time. This means that [latex]A[/latex] and [latex]B[/latex] do not share any outcomes and [latex]P(A \text{ AND } B) = 0[/latex]. For example, suppose the sample space [latex]S[/latex] = {[latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10[/latex]}. Let [latex]A[/latex] = {[latex]1, 2, 3, 4, 5[/latex]}, [latex]B[/latex] = {[latex]4, 5, 6, 7, 8[/latex]}, and [latex]C[/latex] = {[latex]7, 9[/latex]}. [latex]A \text{ AND } B[/latex] = {[latex]4, 5[/latex]}. [latex]\displaystyle{P}{({A} \text{ AND } {B})}=\frac{{2}}{{10}}[/latex]and is not equal to zero. Therefore, [latex]A[/latex] and [latex]B[/latex] are not mutually exclusive. [latex]A[/latex]and [latex]C[/latex] do not have any numbers in common so [latex]P(A \text{ AND } C) = 0[/latex]. Therefore, [latex]A[/latex] and [latex]C[/latex] are mutually exclusive. If it is not known whether [latex]A[/latex] and [latex]B[/latex] are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms.
Flip two fair coins. (This is an experiment.) The sample space is {[latex]HH[/latex], [latex]HT[/latex], [latex]TH[/latex], [latex]TT[/latex]} where [latex]T[/latex] = tails and [latex]H[/latex] = heads. The outcomes are [latex]HH[/latex], [latex]HT[/latex], [latex]TH[/latex], and [latex]TT[/latex]. The outcomes HT and TH are different. The [latex]HT[/latex] means that the first coin showed heads and the second coin showed tails. The [latex]TH[/latex] means that the first coin showed tails and the second coin showed heads.
Draw two cards from a standard [latex]52[/latex]-card deck with replacement. Find the probability of getting at least one black card.
Flip two fair coins. Find the probabilities of the events.
This video provides two more examples of finding the probability of events that are mutually exclusive.
A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events:
Roll one fair, six-sided die. The sample space is {[latex]1, 2, 3, 4, 5, 6[/latex]}. Let event
Are [latex]C[/latex] and [latex]E[/latex] mutually exclusive events? (Answer yes or no.) Why or why not?
Let event [latex]A[/latex] = learning Spanish. Let event [latex]B[/latex] = learning German. Then [latex]A[/latex] AND [latex]B[/latex] = learning Spanish and German. Suppose [latex]P(A) = 0.4[/latex] and [latex]P(B) = 0.2[/latex]. [latex]P(A \text{ AND } B) = 0.08[/latex]. Are events [latex]A[/latex] and [latex]B[/latex] independent? Hint: You must show ONE of the following:
Let event [latex]G[/latex] = taking a math class. Let event [latex]H[/latex] = taking a science class. Then, [latex]G[/latex] AND [latex]H[/latex] = taking a math class and a science class. Suppose [latex]P(G) = 0.6[/latex], [latex]P(H) = 0.5[/latex], and [latex]P(G \text{ AND } H) = 0.3[/latex]. Are [latex]G[/latex] and [latex]H[/latex] independent? If [latex]G[/latex] and [latex]H[/latex] are independent, then you must show ONE of the following:
Note: The choice you make depends on the information you have. You could choose any of the methods here because you have the necessary information.
Since [latex]G[/latex] and [latex]H[/latex] are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent (that is, they are dependent) then knowing that a person is taking a science class would change the chance he or she is taking math. For practice, show that [latex]P(H|G) = P(H)[/latex] to show that [latex]G[/latex] and [latex]H[/latex] are independent events.
In a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers [latex]1, 2, 3, 4, 5[/latex], and [latex]6[/latex]. The green marbles are marked with the numbers [latex]1, 2, 3[/latex], and [latex]4[/latex].
[latex]S[/latex] has ten outcomes. What is [latex]P(G \text{ AND } O)[/latex]?
Let event [latex]C[/latex] = taking an English class. Let event [latex]D[/latex] = taking a speech class. Suppose [latex]P(C) = 0.75[/latex], [latex]P(D) = 0.3[/latex], [latex]P(C|D) = 0.75[/latex] and [latex]P(C \text{ AND } D) = 0.225[/latex]. Justify your answers to the following questions numerically.
A student goes to the library. Let events [latex]B[/latex] = the student checks out a book and [latex]D[/latex] = the student checks out a DVD. Suppose that [latex]P(B) = 0.40[/latex], [latex]P(D) = 0.30[/latex] and [latex]P(B \text{ AND } D) = 0.20[/latex].
In a box there are three red cards and five blue cards. The red cards are marked with the numbers [latex]1, 2[/latex], and [latex]3[/latex], and the blue cards are marked with the numbers [latex]1, 2, 3, 4[/latex], and [latex]5[/latex]. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card. Let [latex]R[/latex] = red card is drawn, [latex]B[/latex] = blue card is drawn, [latex]E[/latex] = even-numbered card is drawn. The sample space [latex]S = R1, R2, R3, B1, B2, B3, B4, B5[/latex]. [latex]S[/latex] has eight outcomes.
In a basketball arena,
Let [latex]A[/latex] be the event that a fan is rooting for the away team. Let [latex]B[/latex] be the event that a fan is wearing blue. Are the events of rooting for the away team and wearing blue independent? Are they mutually exclusive?
In a particular college class, [latex]60[/latex]% of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, [latex]75[/latex]% have long hair. Let [latex]F[/latex] be the event that a student is female. Let [latex]L[/latex] be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?
Note:The choice you make depends on the information you have. You could use the first or last condition on the list for this example. You do not know [latex]P(F|L)[/latex] yet, so you cannot use the second condition.
Mark is deciding which route to take to work. His choices are [latex]I[/latex] = the Interstate and [latex]F[/latex] = Fifth Street.
What is the probability of [latex]P(I \text{ OR } F)[/latex]?
A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let [latex]T[/latex] be the event of getting the white ball twice, [latex]F[/latex] the event of picking the white ball first, [latex]S[/latex] the event of picking the white ball in the second drawing.
ReferencesLopez, Shane, Preety Sidhu. “U.S. Teachers Love Their Lives, but Struggle in the Workplace.” Gallup Wellbeing, 2013. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx (accessed May 2, 2013). Data from Gallup. Available online at www.gallup.com/ (accessed May 2, 2013). Concept ReviewTwo events [latex]A[/latex] and [latex]B[/latex] are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other. Formula ReviewIf [latex]A[/latex] and [latex]B[/latex] are independent, [latex]P(A \text{ AND } B) = P(A)P(B)[/latex], [latex]P(A|B) = P(A)[/latex] and [latex]P(B|A) = P(B)[/latex]. If [latex]A[/latex] and [latex]B[/latex] are mutually exclusive, [latex]P(A \text{ OR } B) = P(A) + P(B)[/latex] and [latex]P(A \text{ AND } B) = 0[/latex]. |