Probability · Lesson 2 of 2
Chapter Summary and Practice
“Probability is just the mathematical study of how likely you are to blame bad luck for a decision you made with full confidence.”
• By the end of this lesson, you should be able to. • Recall all major probability definitions and results quickly. • Choose between direct counting and the complement rule. • Solve mixed questions involving coins, dice, cards and random selection. • Check whether a proposed numerical answer can be a valid probability.
Probability becomes easier when you separate every question into two counts: all equally likely outcomes and the outcomes that make the required event happen. Most questions in this chapter are applications of that idea together with the complement rule.
Theoretical Probability
For equally likely outcomes, probability is the fraction of all possible outcomes that are favourable to the event.
Elementary Events
An elementary event has exactly one outcome. The probabilities of all elementary events in an experiment add up to 1.
Complementary Events
The complement of E is the event that E does not happen.
Impossible, Certain and Valid Probabilities
| Result | Meaning |
|---|---|
| P(E)=0 | Impossible event |
| P(E)=1 | Certain event |
| 0≤P(E)≤1 | Every valid probability lies in this interval |
Useful Counting Facts
| Experiment | Equally likely outcomes |
|---|---|
| One fair coin | 2 |
| One fair die | 6 |
| Two different fair coins | 4 |
| Two different fair dice | 36 |
| One card from a standard deck | 52 |
How to Choose the Right Method
| Question type | Best first move |
|---|---|
| Simple event | Count favourable and total outcomes directly |
| 'Not E' | Use 1-P(E) if E is easier |
| 'At least one' | Often calculate 1-P(none) |
| Two coins/two dice | List ordered outcomes or use a table |
| Card question | Recall the 52-card deck structure |
| Proposed probability outside 0 to 1 | Reject it immediately |
• Assuming events are equally likely without checking. • Treating HT and TH as identical for distinguishable coins. • Assuming all sums of two dice are equally likely. • Confusing 'not green' with one specific other colour. • Accepting a negative probability or a probability above 1.
Guided Practice
Problem
If P(E)=0.18, find P(not E).
- 1.E and not E are complementary.
- 2.P(not E)=1-P(E).
- 3.=1-0.18=0.82.
- 4.Therefore, P(not E)=0.82.
Problem
A bag contains 5 red balls and 7 black balls. Find the probability that a random ball is not red.
- 1.Total balls=12.
- 2.Not red means black, so favourable outcomes=7.
- 3.P(not red)=7/12.
- 4.Check: 1-P(red)=1-5/12=7/12.
Problem
One card is drawn from a well-shuffled deck. Find the probability of a red face card.
- 1.Total outcomes=52.
- 2.Face cards are J, Q and K.
- 3.There are 3 face cards in each red suit and 2 red suits.
- 4.Red face cards=6.
- 5.P(red face card)=6/52=3/26.
Problem
A fair coin is tossed three times. Find the probability that all three results are the same.
- 1.Total ordered outcomes=2×2×2=8.
- 2.They are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
- 3.All same occurs only for HHH and TTT.
- 4.P(all same)=2/8=1/4.
Problem
Two different fair dice are thrown. Find the probability that their sum is 7.
- 1.Total ordered outcomes=36.
- 2.Favourable pairs: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
- 3.There are 6 favourable outcomes.
- 4.P(sum=7)=6/36=1/6.
Quiz
Which value cannot be a probability?
If P(E)=0.73, what is P(not E)?
How many equally likely ordered outcomes are there when two different fair dice are thrown?
A certain event has probability:
For two different fair coins, which set gives the equally likely outcomes?
Make sure you can: • identify equally likely outcomes; • count favourable outcomes correctly; • use the complement rule confidently; • handle coin, die, card and selection questions; • reject any proposed probability outside [0,1].
Practice Problems
- Complete: (i) P(E)+P(not E)=___. (ii) Probability of an impossible event=___. (iii) Probability of a certain event=___. (iv) 0≤P(E)≤___.
- Which cannot be a probability: 3/5, -0.2, 18%, 0.91? Give a reason.
- If P(E)=0.08, find P(not E).
- A bag contains only mango-flavoured sweets. One sweet is selected at random. Find P(orange-flavoured) and P(mango-flavoured).
- The probability that two students have different birthdays is 0.995. Find the probability that they have the same birthday.
- A bag contains 4 red balls and 7 black balls. Find P(red) and P(not red).
- A box contains 6 red, 9 white and 5 green marbles. Find P(red), P(white) and P(not green).
- A piggy bank contains 90 fifty-paise coins, 45 one-rupee coins, 30 two-rupee coins and 15 five-rupee coins. Find P(50p) and P(not ₹5).
- A spinner has 8 equal sectors numbered 1 to 8. Find P(8), P(odd), P(number>2) and P(number<9).
- A fair die is thrown once. Find P(prime), P(number strictly between 2 and 6), and P(odd).
- One card is drawn from a well-shuffled deck. Find P(red king), P(face card), P(red face card), P(jack of spades), and P(diamond).
- Eight defective pens are mixed with 112 good pens. One pen is selected at random. Find P(good).
- A box contains discs numbered 1 to 80. Find P(two-digit), P(perfect square), and P(divisible by 5).
- A fair coin is tossed three times. A player wins if all three results are identical. Find the probability that the player loses.
- A fair die is thrown twice. Find the probability that (i) 6 appears in neither throw, (ii) 6 appears at least once.
Key Takeaways
• Theoretical probability compares favourable outcomes with all equally likely outcomes. • P(E) always lies from 0 to 1. • P(E)=0 means impossible; P(E)=1 means certain. • Complementary probabilities add to 1. • Careful counting—not complicated algebra—is the main skill in this chapter.
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