In a large vase, there are 8 roses,
5 daisies, 12 lilies, and 9 orchids. If 4 flowers are selected
at random, and not replaced, find the probability that at
least 1 of the flowers is a rose. Would you consider this
event likely to occur? Explain your answer.
consider the case where you pick only roses
Prob( all roses) = C(8,4)/C(34,4)
= 70/46376
= 35/23188
So the prob of selecting at least 1 rose = 1 - 35/23188
= 23153/23188 or appr .9985
Highly likely to happen.
seems to me you want 1 - prob(no roses)
rather than 1 - prob(all roses)
To find the probability of at least 1 rose being selected, we need to calculate the probability of the complement event (no roses being selected), and subtract that from 1.
First, let's calculate the probability of not selecting a rose in the first pick. There are 34 flowers total (8 roses + 5 daisies + 12 lilies + 9 orchids), so the probability of not selecting a rose in the first pick is (34 - 8) / 34.
Now, we move on to the second pick. Since the first flower was not replaced, there are now 33 flowers remaining, and the number of roses has decreased by 1. So, the probability of not selecting a rose in the second pick is (33 - 7) / 33.
Similarly, we can calculate the probability for the third and fourth picks. The probability of not selecting a rose in the third pick is (32 - 6) / 32, and for the fourth pick, it is (31 - 5) / 31.
To find the probability of the complement event (no roses being selected in any of the four picks), we multiply these probabilities together:
[(34 - 8) / 34] * [(33 - 7) / 33] * [(32 - 6) / 32] * [(31 - 5) / 31].
Finally, to find the probability of at least 1 rose being selected, we subtract the probability of the complement event from 1:
1 - [(34 - 8) / 34] * [(33 - 7) / 33] * [(32 - 6) / 32] * [(31 - 5) / 31].
As for whether this event is likely to occur, we would need to compare the probability we calculated to some threshold. Unfortunately, I don't have enough information to determine a specific threshold to consider this event likely or not. However, by following the above steps, you can calculate the probability and make a judgment based on your own threshold.