There are 5 brown horses and 4 tan horses in a barn. Sonia will randomly select two horses to ride with her friend. What is the probability that the first horse selected is tan and the second horse selected is brown?
[tex]$\frac{2}{9}$[/tex]
[tex]$\frac{5}{18}$[/tex]
[tex]$\frac{20}{81}$[/tex]
[tex]$\frac{1}{20}$[/tex]
Answer (2)
Calculate the probability of selecting a tan horse first: 9 4 .
Calculate the probability of selecting a brown horse second, given a tan horse was selected first: 8 5 .
Multiply the two probabilities: 9 4 × 8 5 = 72 20 .
Simplify the fraction: 72 20 = 18 5 .
Explanation
Understand the problem and provided data We have a total of 5 brown horses and 4 tan horses, making a total of 9 horses in the barn. Sonia is selecting two horses at random. We want to find the probability that the first horse selected is tan and the second horse selected is brown.
Calculate the probability of selecting a tan horse first First, let's find the probability of selecting a tan horse first. There are 4 tan horses out of a total of 9 horses. So, the probability of selecting a tan horse first is: P ( T a n f i rs t ) = 9 4
Calculate the probability of selecting a brown horse second, given a tan horse was selected first Now, given that a tan horse has already been selected, there are now 8 horses remaining in the barn, of which 5 are brown. So, the probability of selecting a brown horse second, given that a tan horse was selected first, is: P ( B ro w n seco n d ∣ T a n f i rs t ) = 8 5
Calculate the overall probability To find the overall probability of selecting a tan horse first and then a brown horse second, we multiply the two probabilities: P ( T a n f i rs t ∩ B ro w n seco n d ) = P ( T a n f i rs t ) × P ( B ro w n seco n d ∣ T a n f i rs t ) = 9 4 × 8 5
Simplify the fraction Now, let's simplify the fraction: 9 4 × 8 5 = 9 × 8 4 × 5 = 72 20 = 18 5
State the final answer Therefore, the probability that the first horse selected is tan and the second horse selected is brown is 18 5 .
Examples
This type of probability problem can be used in many real-life scenarios. For example, imagine you have a bag of different colored marbles, and you want to know the probability of picking a red marble first and then a blue marble. This is the same type of calculation we just performed. Knowing how to calculate these probabilities can help in games, decision-making, and understanding statistics.
The probability that Sonia selects a tan horse first and then a brown horse is 18 5 . This is calculated by finding the probability of selecting a tan horse first and then the probability of selecting a brown horse second, and multiplying these probabilities. Therefore, the final answer is 18 5 .
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