Final answer:
The probability of drawing two gray marbles in sequence without replacement from a hat containing multiple colored marbles is derived by multiplying the probability of the first draw by the probability of the second draw, which gives \(\frac{10}{153}\), not matching the provided options.
Step-by-step explanation:
To find the probability of drawing a gray marble, not replacing it, and then drawing another gray marble from a hat containing 5 gray, 4 red, 5 white, 2 green, and 2 navy marbles, we need to calculate the probability of two dependent events occurring in sequence. First, we need to determine the probability of drawing one gray marble. Since there are 5 gray marbles out of 18 total marbles (5+4+5+2+2), the probability of drawing a gray marble on the first draw is \(\frac{5}{18}\). Once a gray marble has been drawn, there are now 4 gray marbles left and 17 total marbles remaining. The probability of drawing a second gray marble is therefore \(\frac{4}{17}\).
To find the combined probability of both events, we multiply the probabilities of each event: \(\frac{5}{18} \times \frac{4}{17} = \frac{20}{306}\). Simplifying the fraction, we get \(\frac{10}{153}\), which is not one of the provided options. Therefore, there is likely a mistake in the listed options or in the calculation of the probability.