Bernoulli Trials and Binomial Distribution

A marksman hits $75\%$ of all his targets. What is the probability that he will hit exactly four of his next ten shots?

Fifteen coupons are numbered $1, 2, 3,..., 15,$ respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is $9,$ is:

The survey is conducted in a large factory. If $27\%$ of the factory workers weigh less than $65$ kg and that $25\%$ of the factory workers weigh more than $96$ kg.  [ use the standard values $P(Z<-0.6128\dots )=0.27\text{ and }P(Z<0.6744\dots )=0.75$]

Then the assumed weights of the factory workers is modelled by a normal distribution with mean $\mu$ and standard deviation $\sigma$.

Determine two simultaneous linear equations satisfied by $\mu$ and $\sigma$.

Find the values of $\mu$ and $\sigma$.

There are $12$ white and $12$ red balls in a bag. Balls are drawn one by one with replacement from the bag. The probability that $7^{\mathrm{th}}$ drawn ball is $4^{\mathrm{th}}$ white is

One hundred identical coins, each with probability $p$, of showing up a head, are tossed. If $0<p<1$, and if the probability of heads on exactly $50$ coin is equal to that of heads on exactly $51$ coins, then the value of $p$, is

The mean and variance of a binomial distribution are $5$ and $4$ respectively. Then what is $P(X=1)?$

If the sum of mean and variance of a binomial distribution is $15$ and the sum of their squares is $117$, then the number of independent trials is equal to

The mean and variance of a binomial distribution are $4$ and $3$respectively, then the probability of getting exactly six successes in this distribution is

A boy is throwing stones at a target. The probability of hitting the target at any trial is  $\frac{1}{2}$. The probability of hitting the target for the  $5^{\mathrm{th}}$ time at the ${10}^{\mathrm{th}}$ throw, is

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:

An experiment succeeds twice as often as it fails. The probability of at least $5$, successes in the six trials of this experiment is:

Calcair buys a new passenger jet with $538$ seats. For the first flight of the new jet all $538$ tickets are sold. Assume that the probability that an individual passenger turns up to the airport in time to take their seat on the jet is $0.91$. Let random variable $T=$the number of passengers that arrive on time to take their seats, stating any assumptions you make.

Calcair knows that it is highly likely that there will be some empty seats on any flight unless it sells more tickets than seats. Find the smallest possible number of tick that  P T ≥ 510  is at least  0 . 1 .ets sold so

In Victorian England, the probability of a child born being male was $0.512$. In a family of $10$ children, find the probability there were more girls than boys.

In Victorian England, the probability of a child born being male was $0.512$. In a family of $10$ children, find the probability that there were no boys.

In Victorian England, the probability of a child born being male was $0.512$. In a family of $10$ children, find the probability that there were exactly $6$ boys.

The speeds of cars passing a point on a highway are analysed by the police force. It is found that the speeds follow a normal distribution with mean $115.7 \mathrm{km}/\mathrm{h}$ and standard deviation $10 \mathrm{km}/\mathrm{h}$.

A sample of eight cars is taken. Find the probability that in the sample of eight, more than five cars are travelling between $110 \mathrm{km}/\mathrm{h}$ and $120 \mathrm{km}/\mathrm{h}$. State the assumptions you must make.

An electronics company produces batteries with a lifespan that is normally distributed with a mean of $182$ days and a standard deviation of $10$ days.

In a sample of seven batteries chosen for a quality control inspection, find the probability that no more than three of them last longer than $190$ days.

The number of boys in $90$ families with three children is shown in the table.

If the probability of having a boy is $0.5$, use the binomial expansion $B\left(3,0.5\right)$ to find the expected probabilities.