Question

‘Vehicles approaching a certain road junction from town A can either turn left, turn right or go straight on. Over time it has been noted that of the vehicles approaching this particular junction from town A, 55% turn left, 15% turn right and 30% go straight on. The direction a vehicle takes at the junction is independent of the direction any other vehicle takes at the junction.

(i) Find the probability that, of the next three vehicles approaching the junction from town A, one goes straight on and the other two either both turn left or both turn right. ’

Answer 1

The probability that, out of the next three vehicles at the road junction from town A, one goes straight and the other two either both turn left or both turn right is 9.75%.

Step-by-step explanation:

The question involves calculating the probability that, of the next three vehicles approaching the road junction from town A, one goes straight on and the other two either both turn left or both turn right.

Given the probabilities: P(left) = 55% or 0.55, P(right) = 15% or 0.15, and P(straight) = 30% or 0.30.

To find the required probability, we should consider the two separate scenarios:

1. One vehicle goes straight, and two vehicles turn left.
2. One vehicle goes straight, and two vehicles turn right.

We can use the multiplication rule of independent events to calculate these probabilities:

P(one straight and two left) = P(straight) × P(left) × P(left) = 0.30 × 0.55 × 0.55

P(one straight and two right) = P(straight) × P(right) × P(right) = 0.30 × 0.15 × 0.15

Then, the total probability is the sum of these two probabilities:

P(total) = P(one straight and two left) + P(one straight and two right)

Calculating these:

P(one straight and two left) = 0.30 × 0.55 × 0.55 = 0.09075

P(one straight and two right) = 0.30 × 0.15 × 0.15 = 0.00675

P(total) = 0.09075 + 0.00675 = 0.0975

Therefore, there is a 9.75% chance that of the next three vehicles approaching the junction from town A, one goes straight and the others either both turn left or both turn right.

Answer 2

The probability that out of three vehicles approaching from town A, one goes straight on and the other two either both turn left or both turn right, is 9.75%.

Step-by-step explanation:

To find the probability that, of the next three vehicles approaching the junction from town A, one goes straight on and the other two either both turn left or both turn right, we need to consider the independent events for each vehicle and the combined probability for the three vehicles.

The probabilities given are 55% (or 0.55) for turning left, 15% (or 0.15) for turning right, and 30% (or 0.30) for going straight on. Since we want one vehicle to go straight and the other two to take the same direction (either both left or both right), we can calculate the combined probability by considering the two scenarios: both turning left and both turning right.

Scenario 1: One goes straight on, and two turn left

The probability of one vehicle going straight on is P(Straight) = 0.30.

The probability of two vehicles both turning left is P(Left, Left) = 0.55 * 0.55 = 0.3025.

Combined probability for this scenario is P(Straight, Left, Left) = P(Straight) * P(Left, Left) = 0.30 * 0.3025.

Scenario 2: One goes straight on, and two turn right

The probability of one vehicle going straight on is P(Straight) = 0.30.

The probability of two vehicles both turning right is P(Right, Right) = 0.15 * 0.15 = 0.0225.

Combined probability for this scenario is P(Straight, Right, Right) = P(Straight) * P(Right, Right) = 0.30 * 0.0225.

Since the two scenarios are mutually exclusive, we can add the probabilities of both scenarios to find the total probability of the specific arrangement happening:

Total probability = P(Straight, Left, Left) + P(Straight, Right, Right) = (0.30 * 0.3025) + (0.30 * 0.0225).

Now we calculate the actual probabilities:

P(Straight, Left, Left) = 0.30 * 0.3025 = 0.09075

P(Straight, Right, Right) = 0.30 * 0.0225 = 0.00675

Total probability = 0.09075 + 0.00675 = 0.0975 or 9.75%