1) From the total of 17 marbles, Jacob takes 2 out at random. There are
possible outcomes of the draw. (Lots of different notations are used for the binomial coefficient; the first one is my preference.) The number of ways to draw exactly 2 orange marbles is
That is, of the 3 red and 4 blue marbles - or the 7 non-orange marbles - we want 0; of the 10 oranges, we want 2.
So the probability of drawing exactly 2 orange marbles is .
2) Now Jacob draws 3 marbles, for which there are
possible combinations. The event that Jacob draws at least 1 orange marble is complementary to the event that Jacob draws 0 orange marbles. So if we find the probabilty of drawing 0 oranges, then we subtract this from 1 to find the probability of drawing at least 1.
There are
possible ways of doing, so the probability of drawing 0 oranges is , in turn making the probability of drawing at least 1,
.
3) This question is kinda ambiguous. It's not clear whether Jacob draws a marble with or without replacement.
If he does return a drawn marble to the bag, then for each draw, there is a probability that he draws an orange marble, and a
probability of not. The draws are presumably independent of one another, so that the probability of drawing the first orange marble on the fourth attempt would be
If no replacements are made, then as non-orange marbles are drawn from the bag, the number of non-oranges and the total number of marbles both decrease by 1. Then the probability of drawing orange on the fourth draw would be
If you're supposed to round to the nearest hundredths place, your final answer should be acceptable either way.
4) We want the probability that a boat is made of wood given that it has chrome accents:
5) We apply the inclusion/exclusion principle:
6) The events of a boat having chrome accents and a boat being made of wood are independent if and only if
But , so these two events are not independent.
