How do you solve probability questions on coins, dice and cards?

Before a cricket match, the umpire tosses a coin. There are two equally likely outcomes, heads or tails, and one of them is heads. So the chance of heads is 1 out of 2, or 1/2.

That is the whole idea: count the ways you want, divide by all the possible ways. Probability is always between 0 (impossible) and 1 (certain). Most mistakes come from counting the total outcomes wrongly, so the main skill is listing them carefully.

Coins: Toss 2 coins. Outcomes: HH, HT, TH, TT, so 4 in total.

• At least one head: HH, HT, TH = 3/4.
• Toss 3 coins (8 outcomes). Exactly two heads: HHT, HTH, THH = 3/8.

Dice: Throw 2 dice. There are 6 x 6 = 36 outcomes.

• Sum is 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6/36 = 1/6.
• Sum is 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5/36.
• At least one six: 1 - P(no six) = 1 - (5/6 x 5/6) = 1 - 25/36 = 11/36.

Cards: A deck has 52 cards, 4 suits of 13. Hearts and diamonds are red (26 cards), and there are 12 face cards (J, Q, K of each suit).

• A king: 4/52 = 1/13.
• A face card: 12/52 = 3/13.
• A red card or a king: 26 red + 4 kings - 2 red kings (counted twice) = 28, so 28/52 = 7/13.
• Two cards drawn without replacement, both aces: 4/52 x 3/51 = 12/2,652 = 1/221.

let sevens = 0;
for (let a = 1; a <= 6; a++) {
  for (let b = 1; b <= 6; b++) {
    if (a + b === 7) sevens++;
  }
}
console.log(`${sevens}/36`); // Output: 6/36

• Basic formula: P(E) = favourable / total, for equally likely outcomes.
• Complement: P(not E) = 1 - P(E). Use it for "at least one".
• Or (addition rule): P(A or B) = P(A) + P(B) - P(A and B). Subtract the overlap so it is not counted twice.
• And (independent events): P(A and B) = P(A) x P(B). Two separate dice or coin tosses are independent.
• Without replacement: the second draw has one card fewer, so the numbers change: 4/52, then 3/51.
• Using combinations: the chance that 2 balls drawn from a bag of 4 red and 6 blue are both red = 4C2 / 10C2 = 6 / 45 = 2/15.

Probability questions appear in TCS, Infosys, Deloitte, Amazon and analyst role tests. The same ideas sit behind cricket win predictors, weather forecasts ("70% chance of rain"), insurance pricing and A/B testing in product teams. Interviewers for data roles often start with a coin or dice question and then build on it.

• Treating two coin outcomes as HH, HT and TT only (3 outcomes). HT and TH are different, so there are 4.
• Adding probabilities of events that overlap, like "red or king", without subtracting the red kings.
• Forgetting to reduce the deck after a card is drawn without replacement.
• Working out "at least one" by listing every case instead of using 1 minus the chance of none.
• Forgetting that a face card means J, Q and K only, not aces.

• What is the probability that a leap year has 53 Sundays?
• Three dice are thrown. What is the probability that all show the same number?
• A card is drawn from a deck. What is the probability it is a spade or an ace?