4.5 Concepts of probability questions for IB Maths AA

Describe how the inspector should select the sample.

On one day the randomly chosen starting loaf is loaf number 7. Write down the number of the fourth loaf selected.

Find the probability that a particular loaf is selected on that day.

The loaves are baked on trays that each hold exactly 25 loaves, and they leave the oven in the same order every day. Loaves baked in the corner position of a tray are more likely to be overcooked.

Explain why the systematic sample may give a misleading estimate of the proportion of overcooked loaves.

Find the probability that a randomly chosen bolt is defective.

The machine makes $5000$ bolts in a day. Find the expected number of defective bolts made in a day.

The machine is serviced so that the standard deviation of the diameters is reduced to $\sigma$ mm. The mean stays at $10.00$ mm. After the service, $1\mathrm{\%}$ of bolts are defective.

Find the value of $\sigma$.

Some weeks later, the mean diameter has drifted to $10.02$ mm. The standard deviation is still $\sigma$ mm. Find the percentage of bolts that are now defective.

With the mean at $10.02$ mm, a batch of $200$ bolts is checked. Find the probability that more than $5$ bolts in the batch are defective.

A customer requires that at most $4\mathrm{\%}$ of bolts have a diameter of more than $10.1$ mm. The standard deviation remains $\sigma$ mm. Find the greatest possible mean diameter that meets this requirement. Give your answer correct to three decimal places.

A group consists of 4 boys and 5 girls. A team of 3 is chosen from the group.

By considering the number of boys in the team, show that

Explain, using a counting argument, why

where $m,n,r\in {\mathbb{Z}}^{+}$ and $r\le m$, $r\le n$.

By considering the coefficient of $x^r$ in $(1+x{)}^m(1+x{)}^n$, give a second justification of the result in part (b).

By choosing suitable values of $m$ and $r$ in part (b), show that ${\displaystyle \sum _{k=0}^n{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^2=\left(\genfrac{}{}{0px}{}{2n}{n}\right)}$.

Verify the result in part (d) for $n=3$.

The numbers $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$ are called central binomial coefficients. They also appear in the binomial expansion of $(1-4x{)}^{-\frac{1}{2}}$.

Show that the coefficient of $x^n$ in the expansion of $(1-4x{)}^{-\frac{1}{2}}$ is $\left(\genfrac{}{}{0px}{}{2n}{n}\right)$, for $n\in {\mathbb{Z}}^{+}$.

State the values of $x$ for which the expansion of $(1-4x{)}^{-\frac{1}{2}}$ is valid.

By considering $(1-4x{)}^{-\frac{1}{2}}\times (1-4x{)}^{-\frac{1}{2}}$, show that

Verify the result in part (g) for $n=2$.

A fair coin is tossed $2n$ times. Let $p_n$ be the probability of obtaining exactly $n$ heads, and let $p_0=1$.

Show that $p_n={\displaystyle \frac{1}{4^n}}{\displaystyle \left(\genfrac{}{}{0px}{}{2n}{n}\right)}$, and hence show that ${\displaystyle \sum _{k=0}^n{p}_k{p}_{n-k}=1}$.

Find the least value of $n$ for which $p_n<0.1$.

State whether the length of a fish is a discrete or a continuous variable.

Show that the longest fish is an outlier, and that the shortest fish is not.

The longest fish was photographed when it was measured, and it is of a species that is known to grow to more than 60 cm. State, with a reason, whether its length should be kept in the data.

Estimate the number of these 200 fish that are shorter than 27 cm.

The holes in the nets let fish shorter than about 8 cm escape. Explain how this affects the use of these 200 fish to estimate the median length of all the fish in the lake.

The 200 fish are made up of 120 perch, 50 roach and 30 pike. A stratified sample of 40 of these fish, proportional to species, is chosen for further study. Find the number of fish of each species in this sample.

All 200 fish are tagged and returned to the lake. One month later, 150 fish are caught at random from the lake and 12 of them are found to be tagged. Assume that this second catch is a random sample of all the fish in the lake. Estimate the total number of fish in the lake.

State one further assumption needed for the estimate in part (g) to be reliable.

Find the probability that rain is forecast for a randomly chosen day.

Given that rain is forecast, find the probability that it rains.

Given that rain is forecast, find the probability that the day is sunny.

Given that rain is not forecast, find the probability that it rains.

Find the expected number of days in this period for which rain is forecast but it does not rain.

Find the probability that rain is forecast on at least 10 of the 30 days.

Rain is forecast on exactly 10 of the 30 days. Find the probability that it rains on all 10 of these days.

Find $\mathrm{P}(S)$.

Find $\mathrm{P}(T\mid S)$.

Determine whether the events $S$ and $T$ are independent. Justify your answer.