A car rental agency currently has 41 cars available, 28 of which have a GPS navigation system. Two cars are selected at random from these 41 cars. Find the probability that both of these cars have GPS navigation systems

prob = (28/41)(27/40) = 189/410

or

number of ways to choose 2 cars = C(41,2) = 820
number of ways to choose 2 cars with GPS = C(28,2) = 378
prob of picking two GPS cars = 378/820 = 189/410

To find the probability that both of the selected cars have GPS navigation systems, we need to calculate the ratio of favorable outcomes to the total number of possible outcomes.

First, we need to find the total number of ways to choose two cars from the 41 available cars. This can be determined using the combination formula, which is denoted as "n choose r" and calculated as:

C(n, r) = n! / (r!(n-r)!)

where n is the total number of items and r is the number of items being chosen.

In this case, n = 41 (total number of cars) and r = 2 (number of cars being selected). Substituting these values into the formula, we have:

C(41, 2) = 41! / (2!(41-2)!)
= 41! / (2!39!)

Now, we need to find the number of ways to choose two cars that both have GPS navigation systems. Since 28 out of the 41 cars have GPS navigation systems, the number of favorable outcomes is determined using the combination formula as well:

C(28, 2) = 28! / (2!(28-2)!)
= 28! / (2!26!)

The probability can then be calculated as the ratio of the favorable outcomes to the total possible outcomes:

Probability = Favorable outcomes / Total possible outcomes
= C(28, 2) / C(41, 2)
= (28! / (2!26!)) / (41! / (2!39!))

Now, we can calculate this probability.

To solve this problem, we need to determine the ratio of favorable outcomes (i.e., both cars having GPS navigation systems) to the total number of possible outcomes (i.e., choosing any two cars from the 41 available).

First, let's calculate the number of favorable outcomes: there are 28 cars with GPS navigation systems, and we need to choose 2 of them. We can use the combination formula (nCr) to calculate this:

Number of favorable outcomes = nCr = (28 choose 2) = 28! / [(2!)(28-2)!] = (28 * 27) / (2 * 1) = 14 * 27 = 378.

Next, let's determine the total number of possible outcomes: any 2 cars can be chosen from the 41 available cars. We can again use the combination formula to calculate this:

Total number of possible outcomes = nCr = (41 choose 2) = 41! / [(2!)(41-2)!] = (41 * 40) / (2 * 1) = 41 * 20 = 820.

Finally, we can compute the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes = 378 / 820 = 0.46 (rounded to two decimal places).

Therefore, the probability that both of these cars have GPS navigation systems is 0.46 (or 46%).