Bernoulli Trials and Binomial Distribution

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 probability that an event $A$ happens in one trial of an experiment is $0.4$ . Three independent trials of the experiment are performed. The probability that the event $A$ happens at least once 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 $\sigma$.

The national park houses of a large number of tall trees. The heights of the given trees are normally distributed with mean of $45$ metres and a the standard deviation of $9$ metres. A selection of $50$ trees are chosen randomly then find the probability that at least $5$ of these trees are taller than $55$ meters.

Find the expectation of number of heads in $30$ tosses of a fair coin.

The probability that Steve the striker scores in any given football match is $0.28$. Four matches over the course of a season are selected at random.

Find the probability that Steve scores in at least two of the four matches.

(Write answer in decimal form up to $3$ decimal places)

The probability that Steve the striker scores in any given football match is $0.28$. Four matches over the course of a season are selected at random.

Find the probability that Steve scores in exactly one of the four matches.

(Write answer in decimal form up to $3$ decimal places)

The probability that Steve the striker scores in any given football match is $0.28$. Four matches over the course of a season are selected at random.

Find the probability that Steve scores in at least one of the four matches.

(Write answer in decimal form up to $2$ decimal places)

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.

Percy sews three seeds in each of $50$ different pots. The probability that a seed will germinate is $0.75$. The number of seeds that germinated in each pot is shown in the table.

Using the binomial expansion $\mathrm{B}(3,0.75)$, find the expected probabilities of $0,1,2 \text{or} 3$ seeds germinating.

Xsquared Potato Crisps runs a promotion for a week. In $0.01\%$ of the hundreds of thousands of bags produced there are gold tickets for a round-the-world trip. Let $B$ represent the number of bags of crisps opened until a gold ticket is found.

Hence show that the probability distribution function of $B$ is $f\left(b\right)=P\left(B=b\right)=0.0001{\left(0.9999\right)}^{b-1}$.

Determined to win a ticket, Yimo buys $10$ bags of crisps. Find the probability that she finds a gold ticket after opening no more than $10$ bags.

Xsquared Potato Crisps runs a promotion for a week. In $0.01\%$ of the hundreds of thousands of bags produced there are gold tickets for a round-the-world trip. Let $B$ represent the number of bags of crisps opened until a gold ticket is found.

Hence show that the probability distribution function of $B$ is $f\left(b\right)=P\left(B=b\right)=0.0001{\left(0.9999\right)}^{b-1}$.

State the domain of $f\left(b\right)$.

Xsquared Potato Crisps runs a promotion for a week. In $0.01\%$ of the hundreds of thousands of bags produced there are gold tickets for a round-the-world trip. Let $B$ represent the number of bags of crisps opened until a gold ticket is found.

Hence show that the probability distribution function of $B$ is $f\left(b\right)=P\left(B=b\right)=0.0001{\left(0.9999\right)}^{b-1}$.