Free Guide: Probability Questions in Psychotechnic and Dapar Tests

How to solve dice, coin, and bag-of-balls questions without getting confused while counting

Written by Eli Isaak • Phone: 054-5664932 • Ask me on WhatsApp

About Probability Questions

Probability questions in psychotechnic and Dapar tests almost always deal with the same tools: dice, coins, bags of balls, and student classrooms. The mathematical level is low, and the difficulty lies in orderly counting. Many examinees miss results (for example, that the pair {(2,6)} is different from {(6,2)}), or add probabilities instead of multiplying them.

The key to success is to ask three questions before every calculation: What are all the possible outcomes, what are the desired outcomes, and are the events connected by "AND" (multiplication) or "OR" (addition). Below are four core methods for solving the most common types of questions, with two solved examples for each.

The Golden Triangle of Probability Questions:

Probability = Desired Outcomes / Total Possible Outcomes
Probability is always between 0 and 1. "AND" means multiplication, "OR" means addition, and the complement is 1 - P.
1
Basic Probability (Desired Outcomes Divided by Total Outcomes)
The probability of an event is the ratio between the number of desired outcomes and the number of all possible outcomes, when all outcomes are equally likely. The result is always a number between 0 and 1, and it can be written as a fraction, decimal, or percentage. It is always wise to reduce the fraction.
Example A: A Single Die
A fair die is rolled once. What is the probability of getting an even number?
  • Desired outcomes: 2, 4, 6 - three outcomes
  • All possible outcomes: 1, 2, 3, 4, 5, 6 - six outcomes
  • Probability: 3 / 6 = 1/2
Answer: 1/2 (50%)
Example B: Balls in a Bag
A bag contains 5 red balls, 3 blue balls, and 2 green balls. One ball is drawn at random. What is the probability that the ball is blue?
  • Total balls in the bag: 5 + 3 + 2 = 10
  • Blue balls: 3
  • Probability: 3 / 10
Answer: 3/10 (30%)
2
Rolling Two Dice (36 Possible Outcomes)
When rolling two dice, there are 6 × 6 = 36 equally likely possible outcomes. It is important to remember that the pairs are ordered: the outcome (2,6) is different from the outcome (6,2). It is helpful to know the number of ways to get each sum: sum of 2 - one way, 3 - two ways, 4 - three, 5 - four, 6 - five, 7 - six ways (most common), 8 - five, 9 - four, 10 - three, 11 - two, 12 - one way.
Example A: A Given Sum
Two fair dice are rolled. What is the probability that the sum of the numbers obtained is 8?
  • All outcomes: 6 × 6 = 36
  • Pairs whose sum is 8: (2,6), (3,5), (4,4), (5,3), (6,2) - five pairs
  • Probability: 5 / 36
Answer: 5/36
Example B: At Least One Number
Two fair dice are rolled. What is the probability that at least one of them shows the number 6?
  • 6 on the first die: 6 outcomes, 6 on the second die: 6 outcomes
  • The outcome (6,6) was counted twice and therefore is subtracted once
  • Desired outcomes: 6 + 6 - 1 = 11
  • Probability: 11 / 36
Answer: 11/36
3
The Complementary Event (1 Minus the Opposite Probability)
When it is hard to count the desired outcomes, count the unwanted outcomes and subtract from 1: Probability of the event = 1 minus the probability of the complementary event. This method is especially suited for phrasings like "different", "not", and "at least".
Example A: Different Numbers
Two fair dice are rolled. What is the probability of getting two different numbers?
  • Complement: two identical numbers: (1,1), (2,2), ..., (6,6) - six outcomes
  • Probability of the complement: 6 / 36 = 1/6
  • Requested probability: 1 - 1/6 = 5/6
Answer: 5/6
Example B: Even Product
Two fair dice are rolled. What is the probability that the product of the numbers obtained will be even?
  • Complement: odd product, which occurs only when both numbers are odd
  • Each die has 3 odd numbers: 3 × 3 = 9 outcomes
  • Probability of the complement: 9 / 36 = 1/4
  • Requested probability: 1 - 1/4 = 3/4
Answer: 3/4
4
Independent Events ("AND" - Multiplying Probabilities)
When events do not affect each other (repeated rolls, a die and a coin, two people working separately), and the question requires that both this and that happen, multiply the probabilities.
Example A: Coin and Die
A fair coin is tossed and a fair die is rolled simultaneously. What is the probability of getting heads and a prime number?
  • Probability for heads: 1/2
  • Prime numbers on a die: 2, 3, 5, therefore 3/6 = 1/2
  • Multiplication: 1/2 × 1/2 = 1/4
Answer: 1/4
Example B: Two People
Dana scores a basket with a probability of 3/4 and Yuval scores with a probability of 2/3. Each throws once, independently. What is the probability that both will score?
  • Multiplication: 3/4 × 2/3 = 6/12
  • Reduction: 6/12 = 1/2
Answer: 1/2

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