Probability of selecting one card from a standard deck

Three-part: a simple count for face cards, another simple count for red cards, then the addition rule for face-or-red with the six-card overlap correction.

01 Worked Example Video

Every step of Probability of selecting one card from a standard deck explained on video. Pause anywhere; the embed scrolls independently of the page.

YOUTUBE

02 What This Looks Like on ALEKS

ALEKS randomizes the numbers each attempt, but the question shape stays the same. Here are three example versions you might see.

Face / red / face or red

One card is drawn from a standard 52-card deck. Find P(face card OR red card).

Type as a fraction in lowest terms.

P =

King / spade / king or spade

One card is drawn from a 52-card deck. Find P(king OR spade).

Type as a fraction in lowest terms.

P =

Ace / heart / ace or heart

One card is drawn from a 52-card deck. Find P(ace OR heart).

Type as a fraction in lowest terms.

P =

Heads up: Your ALEKS version will use different numbers. The numbers in the practice below are different too — that way you're exercising the move, not memorizing one answer.

03 The Rule

P(A or B) = P(A) + P(B) − P(A and B) the addition rule

Subtract the overlap cards that are BOTH A and B were counted twice

The addition rule: add the two probabilities, subtract the overlap. Without the subtraction, cards that satisfy both events (like red face cards) get counted twice. Sample space for one card = 52. See it live: P(face or red) = 32/52 = 8/13 ≈ 0.6154 in the Law of Large Numbers Simulator — watch a thousand card draws converge on 8/13 →

Common slips

(1) Just added. P(face) + P(red) = 3/13 + 1/2 = 32/26, which is greater than 1 — impossible. That’s the sign you forgot to subtract the overlap. (2) Wrong overlap count. Red face cards = 6 (J, Q, K of hearts AND of diamonds), not 3 or 12. (3) Forgot to reduce. 32/52 → 8/13. ALEKS expects lowest terms.

04 Practice — Step by Step

Practice this problem step by step

One card is drawn from a standard 52-card deck. Find P(face card OR red card).

Step 1

P(face card)

There are 12 face cards (J, Q, K in each of 4 suits). What’s P(face card)? Type as a fraction in lowest terms.

P(face) =

Step 2

P(red card)

Half the deck is red (hearts and diamonds). What’s P(red)? Type as a fraction in lowest terms.

P(red) =

Step 3

P(face AND red) — the overlap

Red face cards are J, Q, K of hearts AND J, Q, K of diamonds. Count them. What’s P(face AND red)? Type as a fraction in lowest terms.

P(both) =

Step 4

Apply the addition rule

P(face OR red) = P(face) + P(red) − P(both) = 12/52 + 26/52 − 6/52. Type as a fraction in lowest terms.

P(face OR red) =

Q2 Q4

Probability of selecting one card from a standard deck

You walked the addition rule end to end.