Independent Events

Q. state and prove addition therom of probability

Q.

Two dice are tossed once. The probability of getting an even number at the first dice or a total of $8$ is

1. $\frac{1}{36}$

2. $\frac{3}{36}$

3. $\frac{11}{36}$

4. $\frac{5}{9}$

Q.

How do you find the probability of three Independent events $?$

Q. The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c respectively. On these subjects, the student has a $75\%$ chance of passing in at least one, a $50\%$ chance of passing in at least two and a $40\%$ chance of passing in exactly two. Which of the following relations are true.

1. 
2. 
3. 
4. 

Q.

If $P\left(A\right)=0.4,P\left(B\right)=x,P\left(AUB\right)=0.7$, and the events $A$ and $B$ are mutually exclusive, then$x=?$

1. $\frac{3}{10}$

2. $\frac{1}{2}$

3. $\frac{2}{5}$

4. $\frac{1}{5}$

Q. A real estate man has eight master keys to open several new homes. Only one master key will open any given home. If 40% of these homes are usually left unlocked, the probability that the real estate man can get into a specific home, if it is given that he selected 3 keys randomly before leaving his office, is equal to:

1. $\frac{5}{8}$
2. $\frac{3}{8}$
3. $\frac{3}{4}$
4. $\frac{9}{40}$

Q.

A purse contains $4$ copper coins and $3$ silver coins, the second purse contains $6$ copper coins and $2$ silver coins. If a coin is drawn out of any purse, then the probability that it is a copper coin is

1. $\frac{4}{7}$

2. $\frac{3}{4}$

3. $\frac{37}{56}$

4. None of these

Q.

The probability of happening an event $A$ is one trial is $0.4$. The probability that the event $A$ happens at least once in three independent trials is

1. $0.936$

2. $0.784$

3. $0.904$

4. $0.216$

Q. The probability that a randomly selected 2-digit number belongs to the set $\{nϵN:(2^n-2\left)\text{is a multiple of 3}\right\}$ is equal to

Q. Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You are going to pull out one marble, record its color, put it back in the box and draw another marble. What is the probability of pulling out a red marble followed by a blue marble?

1. $\frac{8}{81}$
2. $\frac{2}{27}$
3. $\frac{1}{27}$
4. $\frac{1}{9}$

Q.

Find the probability distribution of the number of successes in two tosses of a die, where a succeess is defined as

number greater than 4

six appears on the die.

Q. An electrical system has open-closed switches S₁, S₂ and S₃ as shown in Figure.
The switches operate independently of one another and the current will flow from A and B either if $S_1$ is closed or if both $S_2$ and $S_3$ are closed. If $P\left(S_1\right)=P\left(S_2\right)=P\left(S_3\right)=\frac{1}{2}$, then find the probability that the circuit will work

1. 
2. 
3. 
4. 

Q.

The probability that $A$ speaks truth is $\frac{4}{5}$, while this probability for $B$ is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact

1. $\frac{4}{5}$

2. $\frac{1}{5}$

3. $\frac{7}{20}$

4. $\frac{3}{20}$

Q.

An ordinary cube has four blank faces, one face marked $2$ another marked $3$. Then the probability of obtaining a total of exactly $12$ in $5$ throws is

1. $\frac{5}{1296}$

2. $\frac{5}{1944}$

3. $\frac{5}{2592}$

4. None of these

Q.

The probability that a certain beginner at golf gets a good shot if he uses the correct club is $\frac{1}{3}$ and the probability of a good shot with an incorrect club is $\frac{1}{4}$. In his bag, there are $5$different clubs, only one of which is correct for the shot in question. If he chooses a club at random and takes a stroke, the probability that he gets a good shot is

1. $\frac{2}{3}$

2. $\frac{1}{15}$

3. $\frac{4}{15}$

4. $\frac{7}{15}$

Q.

How do you find the probability of $A$and $B$ $?$

Q.

Let E and F be two independent events. The probability that both E and F happen is 1/12 and the probability that neither E nor F happens is 1/2 .Then,

1. (a) P(E)=1/3, P(F)=1/4
   

2. (b) P(E)=1/2, P(F)=1/6
   

3. (c) P(E)=1/6, P(F)=1/2
   

4. (d) P(E)=1/12, P(F)=1
   

Q. An unbiased coin is tossed. If the outcome is head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well shuffled pack of nine cards numbered $1,2,\cdots 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is :

1. $\frac{19}{36}$
2. $\frac{15}{72}$
3. $\frac{13}{36}$
4. $\frac{19}{72}$

Q. On a toss of two dice, A throws a total of 5. Then the probability that he will throw another 5 before he throws 7 is

1. $\frac{2}{45}$
2. $\frac{2}{5}$
3. $\frac{1}{81}$
4. $\frac{1}{9}$

Q.

There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item?

Q. Two cards are drawn one by one without replacement from a pack of $52$ cards. The probability that the second card is of higher rank than the first card is
(rank in increasing order can be taken from ace to king)

1. $\frac{1}{17}$
2. $\frac{8}{17}$
3. $\frac{1}{2}$
4. $\frac{16}{17}$

Q. Two persons $A,B$ speaks truth with probabilities $0.6$ and $0.7$ respectively. The probability that they will say the same thing while describing a single event is:

1. $0.56$
2. $0.54$
3. $0.38$
4. $0.94$

Q.

A box contains $15$ tickets numbered $1,2,...,15$. Seven tickets are drawn at random one after the other with replacement. The probability that the greatest number on a drawn ticket is $9$ is

1. ${\left(\frac{9}{10}\right)}^6$

2. ${\left(\frac{8}{15}\right)}^7$

3. ${\left(\frac{3}{5}\right)}^7$

4. None of these

Q.

In an examination, the probability of a candidate solving a question is $\frac{1}{2}$. Out of given $5$ questions in the examination, what is the probability that the candidate was able to solve at least $2$ questions?

1. $\frac{1}{64}$

2. $\frac{3}{16}$

3. $\frac{1}{2}$

4. $\frac{13}{16}$

Q.

If four persons independently solve a certain problem correctly with the probabilities $\frac{1}{2},\frac{3}{4},\frac{1}{4}\mathrm{and}\frac{1}{8}$. Then, the probability that the problem is solved correctly by at least one of them is,

1. $\frac{235}{256}$

2. $\frac{21}{256}$

3. $\frac{3}{256}$

4. $\frac{253}{256}$

Q. $A$ and $B$ are independent events. The probability that both $A$ and $B$ occur is $\frac{1}{20}$ and the probability that neither of them occurs is $\frac{3}{5}$. The probability of occurence of $A$ is

1. $\frac{1}{2}$
2. $\frac{1}{10}$
3. $\frac{1}{4}$
4. $\frac{1}{5}$

Q. A bag contains n white and n black balls. Pairs of balls are drawn without replacement until the bag is empty.The probability that each pair consists of one white and one black ball is

1. 
2. 
3. 
4. 

Q.

In a hurdle race, a player has to cross 10 hurdles. The probability that he will clear hurdles is $\frac{5}{6}$. What is the probability that he will knock down fewer than 2 hurdles?

Q. A box contains $N$ coins, $m$ of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is $\frac{1}{2}$, while it is $\frac{2}{3}$ when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. Then the probability that the coin drawn is fair, when first toss head, second toss tail is:

1. $\frac{9m}{8N+m}$
2. $\frac{9m}{8N-m}$
3. $\frac{9m}{8m-N}$
4. $\frac{9m}{8m+N}$

Q.

A die is thrown three times,

E: 4 appears on the third toss, F: 6 and 5 appears respectively on first two tosses