Secondary 5 Conditional Probability Worksheet

5 of the 6 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 5 of the 6 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Conditional Probability

   A hydroponic greenhouse receives 15 seedling trays, and 6 of them carry a root fungus. An inspector examines two trays chosen at random, one after the other, without replacement.

   1. Given that the first tray examined carries the fungus, what is the probability that the second one does too?
   2. Find the probability that both trays carry the fungus.
   3. Find the probability that exactly one of the two trays carries it.
2. Q2Mathematical Expectation

   This question is built around a diagram or a table of values. Open it in the PDF.

3. Q3The Odds For and the Odds Against

   Two teams are trying to reach the final of a robotics tournament.

   1. The odds against team Volt reaching the final are $7:3$. Find the probability that Volt reaches the final.
   2. Team Nimbus reaches the final with probability $0.35$. State the odds for Nimbus reaching the final, in lowest terms.
   3. Which team is more likely to reach the final?
4. Q4The Types of Probability –

   A maintenance team publishes three figures about one wind turbine.

   1. “The blade monitor has 4 channels built to the same specification, one of which is the icing channel, so a randomly triggered channel is the icing channel with probability $\frac{1}{4}$.”
   2. “Over the last 800 operating hours the turbine was stopped 56 times, so the probability that it stops during a given hour is about $0.07$.”
   3. “The chief engineer puts the chance that the new gearbox needs service before spring at $15\%$.”

   Name the type of probability used in each, and say which of the three would be expected to change if a further 800 hours of data were collected.

5. Q5Types of Events –

   At a conservatory, all 300 students play the piano, the violin, or both: 190 play the piano and 160 play the violin. One student is chosen at random.

   1. How many students play both instruments?
   2. Are the events “plays the piano” and “plays the violin” compatible or incompatible? Justify.
   3. Are these two events independent? Support your answer with a calculation.
   4. Describe the complement of “plays the piano” in words and give its probability.
6. Q6Synthesis — drawing on several sheets in this topic

   A repair shop charges a flat $60 diagnostic fee on every tablet brought in. A quarter of the tablets received are water-damaged; the rest are not. A water-damaged tablet turns out to be repairable with probability $0.4$, a tablet that is not water-damaged with probability $0.9$. A repair earns the shop a further $140; a tablet that cannot be repaired is returned with $20 of the diagnostic fee refunded.

   1. Find the probability that a tablet brought in is repairable.
   2. A tablet turns out not to be repairable. Find the probability that it was water-damaged.
   3. State the odds against a tablet being repairable, in lowest terms.
   4. Find the shop's expected revenue per tablet received.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Without replacement: the second draw is a different experiment (Q1)

Everything in Q1 comes out of one formula, read in the direction the question needs:

Part (a) asks for the conditional probability on its own. The word given means you are no longer looking at the original collection — one item has already gone, so both the count of favourable items and the total count drop by one. Halving only the numerator, or only the denominator, is the classic error and it is worth being deliberate about: rewrite the situation in words ("fourteen trays remain, five of them affected") before writing any fraction.

Mathematical expectation (Q2)

The expectation of a random variable is the average of its values, each weighted by how likely it is:

Before computing, check that the given probabilities add to 1 — if they do not, the table is misread and every later number is wrong. Afterwards, sanity-check the value: an expectation must land between the smallest and the largest value in the table, and for a symmetric distribution it lands exactly at the centre.

Part (b) turns each outcome into money, and it does not require a new table. Expectation is linear: if every unit brings in a fixed amount and there is a fixed cost, the expected profit is that amount times the expectation, minus the cost. Rebuilding the whole distribution in dollars gives the same answer and takes five times as long.

Odds are counts of cases, not probabilities (Q3)

This is the sheet's most reliable source of dropped marks, because the notation looks like a fraction and is not one. Odds compare favourable cases to unfavourable cases. A probability compares favourable cases to all cases.

Theoretical, experimental, subjective (Q4)

Three sources, and the question is always which source a stated number came from. Theoretical probability is counted from a model of equiprobable cases, before any trial takes place. Experimental probability is a relative frequency read off real data. Subjective probability is an expert's judgment about a one-off event that cannot be repeated under identical conditions.

Q4 then asks which of the three would change if more data were collected, and only the experimental one would. A theoretical value is fixed by the model; a subjective one changes only if the person changes their mind. The law of large numbers is the sentence worth having ready: as the number of trials grows, the relative frequency settles down towards the theoretical value — it does not reach it, and no finite number of trials ever proves it.

Compatible and independent are two different questions (Q5)

Q5 asks both about the same pair of events, and they are answered by different calculations. Compatible is about whether the two events can happen together at all: they are compatible when their intersection is non-empty. Independent is about whether knowing one changes the chance of the other:

equivalently, when P(B | A) = P(B). To get the overlap in the first place, use the counting identity n(A ∪ B) = n(A) + n(B) − n(A ∩ B), remembering that "every student plays at least one" is what tells you n(A ∪ B) is the whole group.

Q5(d) asks for a complement in words as well as a probability. The complement of "plays the piano" is "does not play the piano" — and in this particular group, where everyone plays at least one instrument, that description simplifies to "plays only the violin". Saying both is what the mark is for: the general description, and what it amounts to here.

Reversing a condition (Q6)

The synthesis question gives you the probability of an outcome within each category and asks for the probability of the category given the outcome. That reversal is the whole difficulty, and it is done in two steps.

The last part of Q6 puts money on the branches. Each outcome carries an amount, so the expected revenue is a fixed part plus each conditional amount weighted by the probability of the branch it sits on. Work out the probability of each outcome first, in a column, then attach the money — doing both at once is how a term goes missing.

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Getting the most out of it

Draw the tree before you write a single fraction

Every conditional question here — the two-stage draw, the reversed condition, the expected revenue — is the same tree with different labels. Branches out of one node must add to 1, which is a free check on your reading of the question, and the probability of a complete path is the product along it.

Name the events with letters, at the top of the page

"Let F be 'the tray carries the fungus'" takes ten seconds and turns the rest of the question into symbols you can manipulate. It also forces you to notice when a question is about a complement, which is the moment most of the reading errors happen.

Use the sum-to-one check on every partition

Both, exactly one, neither. Repairable, not repairable. Each category of the population. Whenever you have listed every possibility once, the probabilities must add to exactly 1 — and when they do not, you have found the error while it is still cheap to fix.

Common questions