If the letters of the word ALGORITHM are arranged at random in a row what is the probability the letters GOR must remain together as a unit?
\( \dfrac{1}{72} \)
Six new employees, two of whom are married to each other, are to be assigned six desks in a row. If the assignment is made randomly, what is the probability that the married couple will have nonadjacent desks?
\( \dfrac{2}{3} \)
Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.
0.556
An experiment consists of rolling a die until a 2 appears.
(i) How many elements of the sample space correspond to the event that the 2 appears on the \(k^{th}\) roll of the die?
(ii) How many elements of the sample space correspond to the event that the 2 appears not later than the \(k^{th}\) roll of the die?
(i) \(5^{k-1}\) elements
(ii) \( \dfrac{5^{k} - 1}{4} \)
A die is loaded so that each odd number is twice as likely to occur as each even number. Find P(G), where G is the event that a number greater than 3 occurs on a single roll.
\( \dfrac{4}{9} \)
In a large metropolitan area, the probabilities are .87, .36, .30 that a family owns a colour TV, a black & white TV, or both kinds of sets. What is the probability that a family owns either one or both kinds of sets?
0.93
If A and B are mutually exclusive events, P(A) = 0.35 and P(B) = 0.45, find:
(a) P(A')
(b) P(B')
(c) P(A ∪ B)
(d) P(A ∩ B)
(e) P(A ∩ B')
(f) P(A' ∩ B')
(a) 0.65
(b) 0.55
(c) 0.8
(d) 0
(e) 0.35
(f) 0.2
A team of medical students assisting surgeries rates them as very complex, complex, routine, simple and very simple with respective probabilities 0.15, 0.20, 0.31, 0.26, 0.08. Find the probabilities that a particular surgery will be rated:
(a) complex or very complex
(b) neither very complex nor very simple
(c) routine or complex
(d) routine or simple
(a) 0.35
(b) 0.77
(c) 0.51
(d) 0.57
Four candidates A, B, C, D apply for the assignment to coach a school cricket team. If A is twice as likely to be selected as B, and B and C are equally likely, while C is twice as likely to be selected as D, determine:
(a) P(C selected)
(b) P(A not selected)
(a) \( \dfrac{2}{9} \)
(b) \( \dfrac{5}{9} \)
One of four persons John, Rita, Aslam or Gurpreet will be promoted next month. You are told John and Gurpreet have equal chance, Rita’s chances are twice John’s, and Aslam’s chances are four times John’s.
(a) Determine P(John promoted)
(b) P(Rita promoted)
(c) P(Aslam promoted)
(d) P(Gurpreet promoted)
(e) If A = {John promoted or Gurpreet promoted}, find P(A).
(a) \( \dfrac{1}{8} \)
(b) \( \dfrac{1}{4} \)
(c) \( \dfrac{1}{2} \)
(d) \( \dfrac{1}{8} \)
(e) \( \dfrac{1}{4} \)
The Venn diagram shows three events A, B, C with probabilities of intersections given. Determine:
(a) P(A)
(b) P(B ∩ C')
(c) P(A ∪ B)
(d) P(A ∩ B')
(e) P(B ∩ C)
(f) Probability that exactly one of the three occurs.
(a) 0.20
(b) 0.17
(c) 0.45
(d) 0.13
(e) 0.15
(f) 0.51
One urn contains two black balls (labelled B1 and B2) and one white ball. A second urn contains one black ball and two white balls (labelled W1 and W2). One of the two urns is chosen at random. Next a ball is randomly drawn from the urn. Then a second ball is chosen at random from the same urn without replacing the first ball.
(a) Write the sample space showing all possible outcomes.
(b) What is the probability that two black balls are chosen?
(c) What is the probability that two balls of opposite colour are chosen?
(a) \( S = \{ B_1B_2, B_1W, B_2B_1, B_2W, WB_1, WB_2, W_1B, W_1W_2, W_2B, W_2W_1 \} \)
(b) \( \dfrac{1}{6} \)
(c) \( \dfrac{2}{3} \)
A bag contains 8 red and 5 white balls. Three balls are drawn at random. Find the probability that
(a) all three balls are white
(b) all three balls are red
(c) one ball is red and two balls are white
(a) \( \dfrac{5}{143} \)
(b) \( \dfrac{28}{143} \)
(c) \( \dfrac{40}{143} \)
If the letters of the word ASSASSINATION are arranged at random, find the probability that
(a) four S’s come consecutively in the word
(b) two I’s and two N’s come together
(c) all A’s are not coming together
(d) no two A’s are coming together
(a) \( \dfrac{2}{143} \)
(b) \( \dfrac{2}{143} \)
(c) \( \dfrac{25}{26} \)
(d) \( \dfrac{15}{26} \)
A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.
\( \dfrac{7}{13} \)
A sample space consists of 9 elementary outcomes \( e_1, e_2, ..., e_9 \) whose probabilities are:
P(e₁) = P(e₂) = .08, P(e₃) = P(e₄) = P(e₅) = .1, P(e₆) = P(e₇) = .2, P(e₈) = P(e₉) = .07.
Suppose A = {e₁, e₅, e₈}, B = {e₂, e₅, e₇, e₉}.
(a) Calculate P(A), P(B), and P(A ∩ B)
(b) Using the addition law of probability, calculate P(A ∪ B)
(c) List the composition of A ∪ B, and calculate P(A ∪ B) by adding probabilities.
(d) Calculate P( B̄ ) from P(B), also calculate P( B̄ ) directly from elementary outcomes.
(a) \( P(A) = 0.25 \), \( P(B) = 0.32 \), \( P(A \cap B) = 0.17 \)
(b) \( P(A \cup B) = 0.40 \)
(c) \( P(A \cup B) = 0.40 \)
(d) \( P(\overline{B}) = 0.68 \)
Determine the probability p, for each of the following events:
(a) An odd number appears in a single toss of a fair die.
(b) At least one head appears in two tosses of a fair coin.
(c) A king, 9 of hearts, or 3 of spades appears in drawing a card from a well-shuffled deck.
(d) The sum of 6 appears in a single toss of a pair of dice.
(a) \( \dfrac{1}{2} \)
(b) \( \dfrac{3}{4} \)
(c) \( \dfrac{3}{26} \)
(d) \( \dfrac{5}{36} \)
In a non-leap year, the probability of having 53 Tuesdays or 53 Wednesdays is
\(\tfrac{1}{7}\)
\(\tfrac{2}{7}\)
\(\tfrac{3}{7}\)
none of these
Three numbers are chosen from 1 to 20. Find the probability that they are not consecutive
\(\tfrac{186}{190}\)
\(\tfrac{187}{190}\)
\(\tfrac{188}{190}\)
\(\tfrac{18}{\binom{20}{3}}\)
While shuffling a pack of 52 playing cards, 2 are accidentally dropped. Find the probability that the missing cards are of different colours
\(\tfrac{29}{52}\)
\(\tfrac{1}{2}\)
\(\tfrac{26}{51}\)
\(\tfrac{27}{51}\)
Seven persons are to be seated in a row. The probability that two particular persons sit next to each other is
\(\tfrac{1}{3}\)
\(\tfrac{1}{6}\)
\(\tfrac{2}{7}\)
\(\tfrac{1}{2}\)
Without repetition of the digits, four-digit numbers are formed with the digits 0, 2, 3, 5. The probability of such a number being divisible by 5 is
\(\tfrac{1}{5}\)
\(\tfrac{4}{5}\)
\(\tfrac{1}{30}\)
\(\tfrac{5}{9}\)
If A and B are mutually exclusive events, then
\(P(A) \le P(\overline{B})\)
\(P(A) \ge P(\overline{B})\)
\(P(A) < P(\overline{B})\)
none of these
If \(P(A \cup B) = P(A \cap B)\) for any two events A and B, then
\(P(A) = P(B)\)
\(P(A) > P(B)\)
\(P(A) < P(B)\)
none of these
Six boys and six girls sit in a row at random. The probability that all the girls sit together is
\(\tfrac{1}{432}\)
\(\tfrac{12}{431}\)
\(\tfrac{1}{132}\)
none of these
A single letter is selected at random from the word "PROBABILITY." The probability that it is a vowel is
\(\tfrac{1}{3}\)
\(\tfrac{4}{11}\)
\(\tfrac{2}{11}\)
\(\tfrac{3}{11}\)
If the probabilities for A to fail is 0.2 and for B is 0.3, then the probability that either A or B fails is
> 0.5
0.5
\(\le 0.5\)
0
The probability that at least one of the events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.2, then \(P(\overline{A}) + P(\overline{B})\) is
0.4
0.8
1.2
1.6
If M and N are any two events, the probability that at least one of them occurs is
\(P(M)+P(N)-2P(M\cap N)\)
\(P(M)+P(N)-P(M\cap N)\)
\(P(M)+P(N)+P(M\cap N)\)
\(P(M)+P(N)+2P(M\cap N)\)
The probability that a person visiting a zoo will see the giraffe is 0.72, the probability that he will see the bears is 0.84 and the probability that he will see both is 0.52.
False
The probability that a student will pass his examination is 0.73, the probability of the student getting a compartment is 0.13, and the probability that the student will either pass or get compartment is 0.96.
False
The probabilities that a typist will make 0, 1, 2, 3, 4, 5 or more mistakes in typing a report are, respectively, 0.12, 0.25, 0.36, 0.14, 0.08, 0.11.
False
If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is .5 and the probability that both A and B are selected is at most .3. Is it possible that the probability of B getting selected is 0.7?
True
The probability of intersection of two events A and B is always less than or equal to those favourable to the event A.
True
The probability of an occurrence of event A is .7 and that of the occurrence of event B is .3 and the probability of occurrence of both is .4.
False
The sum of probabilities of two students getting distinction in their final examinations is 1.2.
True
The probability that the home team will win an upcoming football game is 0.77, the probability that it will tie the game is 0.08, and the probability that it will lose the game is ____.
0.15
If \(e_1, e_2, e_3, e_4\) are the four elementary outcomes in a sample space and \(P(e_1)=0.1,\; P(e_2)=0.5,\; P(e_3)=0.1\), then the probability of \(e_4\) is ____.
0.3
Let \(S=\{1,2,3,4,5,6\}\) and \(E=\{1,3,5\}\), then \(\overline{E}\) is ____.
\(\{2,4,6\}\)
If A and B are two events associated with a random experiment such that \(P(A)=0.3,\; P(B)=0.2\) and \(P(A\cap B)=0.1\), then the value of \(P(A\cap \overline{B})\) is ____.
0.2
The probability of happening of an event A is 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then the probability of neither A nor B is ____.
0.2
Match the proposed probability under Column C₁ with the appropriate written description under Column C₂:
| Column A (C₁) | Column B (C₂) |
|---|---|
0.95 | An incorrect assignment |
0.02 | No chance of happening |
-0.3 | As much chance of happening as not |
0.5 | Very likely to happen |
0 | Very little chance of happening |
Match the following:
| Column A | Column B |
|---|---|
If E₁ and E₂ are mutually exclusive events | E₁ ∩ E₂ = E₁ |
If E₁ and E₂ are mutually exclusive and exhaustive | (E₁ − E₂) ∪ (E₁ ∩ E₂) = E₁ |
If E₁ and E₂ have common outcomes | E₁ ∩ E₂ = ∅, E₁ ∪ E₂ = S |
If E₁ ⊂ E₂ | E₁ ∩ E₂ = ∅ |