Two cards in a row: putting back or keeping out
Draw two cards one after the other and find the chance that both are aces, both hearts or both red: once putting the first card back, once keeping it out. A probability tree shows why the second draw changes.
Core concept
P(A and B) = P(A) × P(B after A): with the card put back the draws are independent; kept out, the second depends on the first
Topic: Probability of combined events
A pack of playing cards has 52 cards in four suits: spades ♠ and clubs ♣ are black, and hearts ♥ and diamonds ♦ are red. Each suit has 13 cards: the ace, the number cards 2 to 10, and three face cards, the jack, the queen and the king. Draw a card, then draw another. If you put the first card back and shuffle, the pack is the same for the second draw: the two draws are independent, and P(both) = P(first) × P(second) with the same probability: How likely something is to happen, from 0 (impossible) to 1 (certain). All key words twice. If you keep the first card out, the pack has only 51 cards, and if the first card was an ace there are only 3 aces left. The second draw depends on the first. A probability tree shows each draw as a branch: multiply along the branches to find the chance of both.
What you will learn
- Draw a probability tree for two cards drawn one after the other.
- Find P(both) for independent draws, with the card put back.
- Find P(both) for dependent draws, with the card kept out, and explain the difference.
Before you start
- Find the probability of a single event when drawing a card.
- Multiply fractions.
Materials
- A pack of 52 playing cards (jokers taken out)
Equipment
- Paper and a pencil for the probability tree
Procedure
- Choose an event and whether the first card is put back.
- Read the two branches of the tree and multiply them to find P(both).
- Compare two aces, two hearts and two red cards with and without putting the first card back.
How this simulation is simplified
- The pack has 52 cards and no jokers, and it is shuffled so every card is equally likely.
- When the first card is put back, the pack is shuffled again before the second draw.