First, we need to find the total number of coins in the jar, which is the sum of all the coins:
8 pennies + 13 dimes + 12 nickels + 22 quarters = 55 coins
The probability of randomly grabbing a nickel first is:
12 nickels / 55 total coins = 12/55
After grabbing a nickel, there are now only 54 coins left in the jar. The probability of randomly grabbing a dime next is:
13 dimes / 54 total coins = 13/54
To find the probability of both events happening, we multiply the probabilities:
(12/55) * (13/54) = 156/2970 = 0.0525
Therefore, the probability of randomly grabbing a nickel and then, without replacement, a dime is 0.0525 or 156/2970.