I was asked to find the probaility if a board game uses the deck of 20 cards. And find how many have the number 3. I came up with 4/20 and was told that I'm on the right track but I need to find P(3 and 3) with replacement and without replacement.
What does this mean?
When we talk about probability, we are referring to the likelihood of an event occurring. In the context of a board game using a deck of 20 cards, you were asked to find the probability of drawing a card with the number 3.
To calculate the probability, you need to know the number of cards with the number 3 and the total number of cards in the deck. In this case, you correctly thought that there are 4 cards with the number 3 out of a total of 20 cards. So, the probability of drawing a card with the number 3 is 4/20.
Now, you're being asked to consider two scenarios: finding the probability with replacement and without replacement.
1. Probability with replacement: When you calculate the probability with replacement, it means that after drawing a card from the deck, you put it back in before drawing the next card. In this case, the probability of drawing the number 3 on the first draw is indeed 4/20. However, even if you draw a card with the number 3 on the first draw, it doesn't affect the probability of drawing another card with the number 3 on the second draw. Therefore, the probability remains the same for subsequent draws as well.
2. Probability without replacement: Conversely, when you calculate the probability without replacement, it means that after drawing a card from the deck, you don't put it back in. Therefore, the number of cards in the deck changes after each draw. In this case, since you don't replace cards, the probability of drawing a card with the number 3 on the first draw is still 4/20. However, after drawing a card with the number 3, the total number of cards in the deck decreases to 19, and the number of cards with the number 3 also decreases to 3. Thus, on the second draw, the probability of drawing a card with the number 3 is 3/19.
To summarize:
- Probability with replacement: P(3) = 4/20 for each draw.
- Probability without replacement: P(3) = 4/20 on the first draw, and for subsequent draws, the probability will be different due to the change in the number of cards in the deck.
Remember, probability represents how likely or unlikely an event is to occur, and it is important to consider whether replacement is being done when calculating probabilities.