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Probability Practice Questions with Detailed Solutions – Level 3

Hard probability problems on conditional events, binomial trials, bag transfers, turn-based draws and multi-card combinations, each with a step-by-step worked solution.

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Questions, answer key and solutions.

Questions
50
Duration
75 min
Marks
50

Negative marking

Attempts
100
Access
Free
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Normal practice, or a Simulated test

Both use the same questions. Normal practice is untimed and needs no account: choose an option and the answer and solution appear at once. The Simulated test is the real computer-based format, with one 75-minute clock, a question palette and full screen, and your score, the key and the solutions arrive when you submit.

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How the two modes differ
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Normal

Untimed

Every question on one page, loading as you scroll. Choose an option and see straight away whether it was right.

  • No timer. Nothing to start and nothing to submit. Stop whenever you like.
  • Answered as you choose. Tap an option and it is marked at once, with the correct answer shown.
  • Solutions in place. The explanation, and the video where one has been added, sit under the question.
  • Open to guests. Normal practice needs no sign-in.
Real exam conditions

Simulated

Timed CBT

The computer-based test format: one clock, a question palette, and a score the moment you submit.

  • A clock that does not pause. 75 minutes from the moment you begin, and the paper is submitted for you when it reaches zero.
  • Question palette. Answered, not answered and marked for review, section by section, as the real interface has it.
  • Full screen, proctored. The test takes over the screen; leaving it three times submits the paper.
  • Answers hidden until you submit. The key, the explanations and your score all arrive together at the end.

About this test

Probability Practice Quiz Level 3: Conditional and Binomial Problems

This Level 3 set checks whether you can handle advanced probability: conditional events, repeated trials, bag transfers and multi-card combinations. It tests whether you can choose the right method rather than just count outcomes. Each problem in this set has a step-by-step worked solution on Aptitude360.

Advanced probability topics in this Level 3 set

  • Repeated coin tosses and die throws using the binomial approach
  • Conditional probability, such as a known dice sum
  • Moving a ball from one bag to another, then drawing from the second bag
  • Three-card draws by suit and rank, and pairs of the same rank or colour
  • Sequential draws until a colour appears, and players taking turns to draw
  • Divisibility by neither of two numbers, and products of drawn numbers
  • Truth-telling problems and the least number of shots for a target probability

Key formulas for hard probability problems

  • Conditional, with \( P(B)>0 \): \( P(A\mid B)=\dfrac{P(A\cap B)}{P(B)} \)
  • Binomial (independent trials, same \( p \)): \( P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k} \)
  • Total probability, where the \( B_{i} \) partition the sample space and each \( P(B_{i})>0 \): \( P(A)=\sum P(B_{i})P(A\mid B_{i}) \)
  • At least one success in \( n \) independent trials, each with probability \( p \): \( 1-(1-p)^{n} \)
  • Largest value exactly \( m \) on \( n \) fair dice: \( \dfrac{m^{n}-(m-1)^{n}}{6^{n}} \)

Revision tips for Level 3 probability

  • In bag-transfer questions, split into cases by the colour moved, then add the weighted results.
  • For a ball drawn at a later position without replacement, symmetry gives the same answer as the first draw.
  • With a given condition, shrink the sample space to the outcomes that satisfy it before counting.

Who should take this hard set

It suits students who have finished basic and medium probability and want exam-level practice on conditional, binomial and multi-stage problems.

FAQs about Probability Practice Questions with Detailed Solutions – Level 3

What is conditional probability?
Conditional probability is the chance of an event given that another event has already happened. It equals the probability of both events divided by the probability of the given event, provided that probability is not zero.
What is binomial probability used for?
It is used for a fixed number of independent trials, each with the same chance of success, such as repeated coin tosses or shots at a target. It gives the probability of an exact number of successes.
What is the theorem of total probability?
If an experiment can go through several mutually exclusive cases that cover all possibilities, each with a non-zero probability, the probability of an event is the sum, over all cases, of each case's probability times the event's probability in that case.
How do you solve probability questions where a ball is transferred from one bag to another?
Split the problem into cases by the colour of the ball transferred. For each case, multiply the chance of that transfer by the chance of the required draw from the updated second bag, then add the results.

Test structure

How the paper is split, and how long each part runs. A section you have finished cannot be returned to.

SectionQuestionsTime
Probability Level 35075 min

Marking & rules

Total marks
50 across 50 questions.
Negative marking
A wrong answer costs marks. Leaving a question unanswered costs nothing.
Attempts
100 in total. An attempt is spent the moment you begin one.

What you get when you submit

The analysis is ready the moment you submit the paper.

Score and accuracy

Marks, percentage, right, wrong and skipped, and the time you took.

Section by section

The same figures for each section, so a weak one cannot hide in an average.

The full answer key

Every question with your answer, the right one, the working, and the video where there is one.

Where you stand

Your score against the spread of everyone else who has sat this test.

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