SAT-58: Probability — Simple and Independent Events

SAT-58: Probability — Simple and Independent Events

Description: Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). This lesson covers single ("simple") events and how to combine independent events, where one outcome does not affect the other.

The basic formula

P(event) = (favorable outcomes) ÷ (total outcomes)

Every probability is between 0 and 1. You can write it as a fraction, decimal, or percent. (Oʻzbekcha: ehtimollik — kerakli natijalar sonini umumiy natijalar soniga boʻlish.)

The complement (the "not" trick)

The probability that an event does not happen is 1 − P(event). This is often the fastest path when "at least" appears in a question. (Oʻzbekcha: hodisa roʻy bermaslik ehtimoli = 1 − P.)

Worked Example 1 — simple event

A bag has 3 red, 2 blue, and 5 green marbles. What is the probability of drawing a blue marble?

• Total = 3 + 2 + 5 = 10. Favorable (blue) = 2.
• P(blue) = 2/10 = 1/5 = 0.2 = 20%.

Worked Example 2 — complement

Using the same bag, what is the probability of not drawing green?

• P(green) = 5/10 = 1/2. So P(not green) = 1 − 1/2 = 1/2.

(Oʻzbekcha: "yashil emas" ehtimoli = 1 − yashil ehtimoli.)

Independent events — multiply

Two events are independent if one does not change the other (e.g. flipping a coin twice, or rolling a die then spinning a spinner). For independent events A and B:

P(A and B) = P(A) × P(B)

(Oʻzbekcha: mustaqil hodisalar uchun ehtimolliklarni koʻpaytiramiz.)

Worked Example 3 — independent events

You flip a fair coin and roll a fair die. What is the probability of heads and a 4?

• P(heads) = 1/2, P(rolling 4) = 1/6.
• P(heads and 4) = 1/2 × 1/6 = 1/12.

Watch the wording: "and" usually means multiply (both happen); for independent events this is exact. (Oʻzbekcha: "va" — ikkalasi ham roʻy berishi → koʻpaytirish.)

Independent vs dependent — the replacement test

Events are independent only if the first does not change the setup for the second. The classic clue is the word replacement. If you draw a marble, put it back, then draw again, the totals reset and the draws are independent — multiply the original probabilities. If you draw and do not replace it, the second draw has fewer items to choose from, so the events are dependent and the second probability changes. Always check whether the item is returned before multiplying. (Oʻzbekcha: agar olingan narsa qaytarib qoʻyilsa — mustaqil; qaytarilmasa — bogʻliq, ikkinchi ehtimol oʻzgaradi.)

Worked Example 4 — dependent (no replacement)

A box has 4 red and 6 blue pens. You draw two pens without replacing the first. What is P(both red)?

• First red: 4/10. After removing one red, 3 reds remain out of 9 pens, so second red: 3/9.
• P(both red) = 4/10 × 3/9 = 12/90 = 2/15.

Notice the totals dropped from 10 to 9 — that is what makes it dependent. (Oʻzbekcha: umumiy son 10 dan 9 ga tushdi — shu sababli hodisalar bogʻliq.)

"At least one" — use the complement

When a question asks for "at least one," computing every case is slow. Instead find the probability of none and subtract from 1: P(at least one) = 1 − P(none). This shortcut saves time on harder SAT questions. (Oʻzbekcha: "kamida bitta" — 1 dan "hech biri yoʻq" ehtimolini ayiramiz.)

Practice 1

A spinner has 8 equal sections numbered 1–8. What is the probability of landing on an even number?

Practice 2

You draw a card and replace it, then draw again from a deck where P(ace) = 1/13 each time. Probability of two aces?

Key words — Kalit soʻzlar

• Probability — ehtimollik
• Event — hodisa
• Outcome — natija
• Favorable outcome — qulay natija
• Complement (not) — toʻldiruvchi (roʻy bermaslik)
• Independent — mustaqil
• Fair (coin/die) — bir jinsli (adolatli)
• With replacement — qaytarib qoʻyish bilan
• Certain / Impossible — muqarrar / mumkin emas

Summary

• P(event) = favorable ÷ total, always between 0 and 1.
• P(not event) = 1 − P(event) (use for "at least" questions).
• Independent events: P(A and B) = P(A) × P(B).