Final answer:
The probability all three cards Chloe turns over are red is found by multiplying the probabilities of drawing a red card in each of the three draws without replacement. The final calculated probability is approximately 0.148, or 14.8%.
Step-by-step explanation:
Chloe is performing a magic trick and wants to know the probability that all three cards she turns over from a set of 10 red cards and 5 black cards are red. To find this probability, we consider each draw one at a time, without replacement. After each draw, the total number of cards decreases, which changes the probability for the next draw.
For the first card, the probability of drawing a red card is 10/15 or 2/3, since there are 10 red cards out of a total of 15. After drawing one red card, there are now 9 red cards left and 14 cards in total, so the probability of drawing a second red card is 9/14. Finally, after drawing two red cards, there are 8 red cards left and 13 cards in total, so the probability of drawing a third red card is 8/13.
To find the overall probability of all three cards being red, we multiply these probabilities together:
- The probability of the first card being red: 10/15 or 2/3
- The probability of the second card being red: 9/14
- The probability of the third card being red: 8/13
Therefore, the combined probability is (2/3) × (9/14) × (8/13), which we can calculate as approximately 0.148, or 14.8%.
It's important to note that this calculation assumes that the cards are not returned to the set after being turned over, as the question indicates that we are drawing without replacement. The process of calculating these probabilities is an example of a dependent event, where the outcome of each draw affects the probability of the next.