In a horse race with 6 horses, you make a bet by predicting the ranking of all 6 horses. If you place your bet at random, what is the probability that you will get the first and second horse correct and in the correct order? Enter your answer as a simplified fraction. Provide your answer below: FEEDBACK SUBMIT
The Correct Answer and Explanation is:
To solve this problem, we are asked to find the probability of getting the first and second horses correct in the correct order in a race with 6 horses.
Step 1: Understanding the Total Number of Possible Outcomes
When you place a bet predicting the ranking of all 6 horses, you are essentially guessing the order in which the horses will finish. There are 6 horses, so the total number of possible ways they can finish is given by the total permutations of 6 objects:6!=6×5×4×3×2×1=7206! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 7206!=6×5×4×3×2×1=720
Thus, there are 720 possible ways the horses can finish in the race.
Step 2: Favorable Outcomes
The favorable outcome is when you get both the first and second horses correct and in the correct order. Once you have correctly identified the first and second horses, there are still 4 remaining horses to be ranked in the remaining 4 positions. So, the number of favorable outcomes is the number of ways to arrange the remaining 4 horses in the last 4 positions, which is:4!=4×3×2×1=244! = 4 \times 3 \times 2 \times 1 = 244!=4×3×2×1=24
Thus, there are 24 favorable outcomes where you correctly predict the first and second horses in the correct order.
Step 3: Calculating the Probability
The probability of getting the first and second horses correct and in the correct order is the ratio of favorable outcomes to the total number of possible outcomes:P(correct first and second)=24720=130P(\text{correct first and second}) = \frac{24}{720} = \frac{1}{30}P(correct first and second)=72024=301
Final Answer:
The probability of getting the first and second horses correct and in the correct order is 130\frac{1}{30}301.
