MEDIUM

Earn 100

Zeke explores his biased coin in more detail. He design a spreadsheet to simulate five throws of his biased (red) coin. The probability of head is $0.964$ . He also throws a fair (black) coin five times. He collects data for $614$ trials as shown below. The image also shows the outcome of the $614 th$ trial.

Let $R$ and $B$ represent the number of heads in five throws of the red coin and five throws of the black coin respectively.

Find $E (R)$ and $E (B)$ and interpret your results.

Important Questions on Probability

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

X ~ B (n, p)

with

n = 10, p = 0.4

then

E (X^{2}) =

EASY

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

The mean and variance of a random variable

X

in binomial distribution are

4

and

2

respectively, then

P (X = 1)

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is

\frac{1}{4}

. If the probability that at most two machines will be out of service on the same day is

{(\frac{3}{4})}^{3} k,

then

k

is equal to

EASY

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

A die is thrown four times. The probability of getting perfect square in at least one throw is

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

An experiment succeeds twice as often as it fails. The probability of at least

5

successes in the six trials of this experiment is

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

Let

C_{1}

and

C_{2}

be two biased coins such that the probabilities of getting head in a single toss are

\frac{2}{3}

and

\frac{1}{3}

, respectively. Suppose

α

is the number of heads that appear when

C_{1}

is tossed twice, independently, and suppose

β

is the number of heads that appear when

C_{2}

is tossed twice, independently. Then the probability that the roots of the quadratic polynomial

x^{2} - α x + β

are real and equal, is

EASY

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

In a binomial distribution, mean is

18

and variance is

12

then probability of success

(p) = . . . .

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

If on an average

9

mountain climbers out of

10

return safely, what is the probability that with

5

climbers out, at least

4

will return safely?

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least

90 %

is:

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is

\frac{4}{5},

then the probability that he is unable to solve less than two problems is

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

A die is thrown

7

times. What is the probability that an even number turns up at least

4

times?

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

A shooter can hit a given target with probability

\frac{1}{4} .

She keeps firing a bullet at the target until she hits it successfully three times, and then she stops firing. The probability that she fires exactly six bullets lies in the interval

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

A coin is tossed

7

times. Each time a man calls head. The probability that he wins the toss at least four times is

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

The probability of guessing correctly at least

8

out of

10

answers on a true-false type examination is

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

The sum of mean and variance of a Binomial distribution

(\vec{B D})

for

3

trials is

2.7

, then the binomial distribution

\vec{B D}

is given by

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

2 n

unbiased coins are tossed. The probability of having the number of heads is not equal to the number of tails is

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

If the mean and standard deviation of a binomial distribution are

20

and

4

, respectively, then the number of trials is ________

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

Let a random variable

X

has a binomial distribution with mean

8

and variance 4. If

P (X \leq 2) = \frac{k}{2^{16}},

then the value of

k

is equal to

MEDIUM

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

For the following distribution function $F (x)$ of a random variable $X$

$x$	$1$	$2$	$3$	$4$	$5$	$6$
$F (x)$	$0.2$	$0.37$	$0.48$	$0.62$	$0.85$	$1$

$P (3 < X \leq 5) =$

HARD

Mathematics>Algebra>Probability>Bernoulli Trials and Binomial Distribution

If the mean and the variance of a binomial variate

X

are

2 & 1

respectively, then the probability that

X

takes a value greater than or equal to one is:

Important Questions on Probability

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