Let's examine Faria's claim step by step to determine whether she is correct.
1. Determine the probability of picking the first green bead:
- According to the information, 60% of the beads in the bag are green.
- Therefore, the probability of picking a green bead on the first draw is:
[tex]\[
\text{Probability of first green bead} = \frac{60}{100} = 0.6
\][/tex]
2. Determine the probability of picking the second green bead:
- After picking the first green bead (assuming we do not replace it), the total number of beads in the bag decreases by one. Also, the number of green beads decreases by one (since one green bead was taken out).
- Originally, there are 6 green beads out of a total of 10 beads.
- After picking one green bead, there are now 5 green beads left out of the remaining 9 beads.
- Thus, the probability of picking a green bead on the second draw is:
[tex]\[
\text{Probability of second green bead} = \frac{5}{9} \approx 0.5555555555555556
\][/tex]
3. Calculate the probability of both beads being green:
- The probability of both events happening (both beads being green) is the product of the individual probabilities:
[tex]\[
\text{Probability of both green beads} = \left(0.6\right) \times \left(\frac{5}{9}\right) = 0.6 \times 0.5555555555555556 \approx 0.3333333333333333
\][/tex]
- Simplifying this result gives:
[tex]\[
\frac{6}{10} \times \frac{5}{9} = \frac{30}{90} = \frac{1}{3}
\][/tex]
4. Compare the calculated probability with Faria's given probability:
- Faria claims that the probability of picking both green beads is [tex]\(\frac{1}{3}\)[/tex].
- Our calculated probability is also [tex]\(\frac{1}{3}\)[/tex].
5. Conclusion:
- Since our calculated probability matches the probability given by Faria, we conclude that Faria is indeed correct.
Thus, the probability calculations confirm Faria's statement that the probability of both beads being green is [tex]\(\frac{1}{3}\)[/tex].
