Then \(\text{A AND B}\) = learning Spanish and German. \(P(\text{A})P(\text{B}) = \left(\dfrac{3}{12}\right)\left(\dfrac{1}{12}\right)\). \(P(\text{E}) = \dfrac{2}{4}\).Let \(\text{F} =\) the event of getting at most one tail (zero or one tail).Let \(\text{G} =\) the event of getting two faces that are the same.Let \(\text{H} =\) the event of getting a head on the first flip followed by a head or tail on the second flip.Are \(\text{F}\) and \(\text{G}\) mutually exclusive?Let \(\text{J} =\) the event of getting all tails. Sampling may be done with replacement or without replacement (Figure \(\PageIndex{1}\)): With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. Let events \(\text{B} =\) the student checks out a book and \(\text{D} =\) the student checks out a DVD. Let \(\text{T}\) be the event of getting the white ball twice, \(\text{F}\) the event of picking the white ball first, \(\text{S}\) the event of picking the white ball in the second drawing.Two events \(\text{A}\) and \(\text{B}\) are independent if the knowledge that one occurred does not affect the chance the other occurs. \(P(\text{J|K}) = 0.3\). You reach into the box (you cannot see into it) and draw one card.The sample space \(S = R1, R2, R3, B1, B2, B3, B4, B5\).Let \(\text{A}\) be the event that a fan is rooting for the away team.Let \(\text{B}\) be the event that a fan is wearing blue.Are the events of rooting for the away team and wearing blue independent? Show that \(P(\text{G|H}) = P(\text{G})\).b. Mutually exclusive event:- two events are mutually exclusive event when they cannot occur at the same time. \(\text{H}\)’s outcomes are \(HH\) and \(HT\).Let \(\text{F} =\) the event of getting the white ball twice.Let \(\text{G} =\) the event of getting two balls of different colors.Let \(\text{H} =\) the event of getting white on the first pick.Are \(\text{G}\) and \(\text{H}\) mutually exclusive?Find the complement of \(\text{A}\), \(\text{A′}\). The first card you pick out of the 52 cards is the \(\text{Q}\) of spades. Let \(\text{H} =\) blue card numbered between one and four, inclusive. One student is picked randomly. \(\text{E} = \{1, 2, 3, 4\}\).Find \(P(\text{C|A})\). Independent Projects and Mutually Exclusive Projects: When acceptance or rejection of a project does not affect the cash flows of other projects, it is known as an independent project whereas when acceptance of a project disqualifies all other contesting projects, it is known as a mutually exclusive … Because you have picked the cards without replacement, you cannot pick the same card twice.You have a fair, well-shuffled deck of 52 cards. The suits are clubs, diamonds, hearts and spades. Since \(\dfrac{2}{8} = \dfrac{1}{4}\), \(P(\text{G}) = P(\text{G|H})\), which means that \(\text{G}\) and \(\text{H}\) are independent.70% of the fans are rooting for the home team.20% of the fans are wearing blue and are rooting for the away team.Of the fans rooting for the away team, 67% are wearing blue.The following probabilities are given in this example:\(P(\text{F}) = 0.60\); \(P(\text{L}) = 0.50\),\(P(\text{I}) = 0.44\) and \(P(\text{F}) = 0.55\).\(P(\text{I AND F}) = 0\) because Mark will take only one route to work.Toss one fair coin (the coin has two sides, \(\text{H}\) and \(\text{T}\)). There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, \(\text{J}\) (jack), \(\text{Q}\) (queen), \(\text{K}\) (king) of that suit.

Therefore, \(\text{A}\) and \(\text{C}\) are mutually exclusive.If it is not known whether \(\text{A}\) and \(\text{B}\) are mutually exclusive.The sample space is \(\{HH, HT, TH, TT\}\) where \(T =\) tails and \(H =\) heads. Let \(\text{L}\) be the event that a student has long hair. Suppose \(P(\text{A}) = 0.4\) and \(P(\text{B}) = 0.2\). Therefore, \(\text{A}\) and \(\text{B}\) are not mutually exclusive. The outcome of the first roll does not change the probability for the outcome of the second roll. (There are five blue cards: \(B1, B2, B3, B4\), and \(B5\). Are they mutually exclusive?So \(P(\text{B})\) does not equal \(P(\text{B|A})\) which means that \(\text{B} and \text{A}\) are not independent (wearing blue and rooting for the away team are not independent). ).Event \(\text{A} =\) heads (\(\text{H}\)) on the coin followed by an even number (2, 4, 6) on the die.Event \(\text{B} =\) heads on the coin followed by a three on the die.

Let \(\text{F}\) be the event that a student is female. Out of the even-numbered cards, to are blue; \(B2\) and \(B4\).

Then \(\text{D} = \{2, 4\}\). \(\text{G} = \{B4, B5\}\). Count the outcomes. It consists of four suits.

P(A AND B) = P(A)P(B) Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are NOT ind… \(P(\text{C AND E}) = \dfrac{1}{6}\).