Answer :
To determine the probability that a student chosen at random from a class has either blonde hair or black hair, we need to use the rule for mutually exclusive events. Specifically, the sum rule states that if two events \( A \) and \( B \) are mutually exclusive, meaning they cannot occur at the same time, then the probability of either event occurring is the sum of their individual probabilities.
Here,
- The probability of a student having blonde hair (\( P(\text{blonde}) \)) is given as \( \frac{5}{13} \).
- The probability of a student having black hair (\( P(\text{black}) \)) is given as \( \frac{7}{26} \).
Since a student cannot have both blonde and black hair simultaneously, these two events are mutually exclusive. Therefore, we add the two probabilities to find the probability that a student has either blonde hair or black hair.
First, let's convert the probabilities to a common denominator:
The probability of having blonde hair is:
[tex]\[ P(\text{blonde}) = \frac{5}{13} \][/tex]
The probability of having black hair is:
[tex]\[ P(\text{black}) = \frac{7}{26} \][/tex]
To combine these, we need a common denominator. Notice that \( 13 \times 2 = 26 \), so we can rewrite \( \frac{5}{13} \) with the denominator \( 26 \):
[tex]\[ \frac{5}{13} = \frac{5 \times 2}{13 \times 2} = \frac{10}{26} \][/tex]
Now we can add the probabilities:
[tex]\[ \text{Probability of either blonde or black hair} = \frac{10}{26} + \frac{7}{26} = \frac{10 + 7}{26} = \frac{17}{26} \][/tex]
Thus, the probability that a student chosen from the class at random has either blonde hair or black hair is:
[tex]\[ \boxed{\frac{17}{26}} \][/tex]
Here,
- The probability of a student having blonde hair (\( P(\text{blonde}) \)) is given as \( \frac{5}{13} \).
- The probability of a student having black hair (\( P(\text{black}) \)) is given as \( \frac{7}{26} \).
Since a student cannot have both blonde and black hair simultaneously, these two events are mutually exclusive. Therefore, we add the two probabilities to find the probability that a student has either blonde hair or black hair.
First, let's convert the probabilities to a common denominator:
The probability of having blonde hair is:
[tex]\[ P(\text{blonde}) = \frac{5}{13} \][/tex]
The probability of having black hair is:
[tex]\[ P(\text{black}) = \frac{7}{26} \][/tex]
To combine these, we need a common denominator. Notice that \( 13 \times 2 = 26 \), so we can rewrite \( \frac{5}{13} \) with the denominator \( 26 \):
[tex]\[ \frac{5}{13} = \frac{5 \times 2}{13 \times 2} = \frac{10}{26} \][/tex]
Now we can add the probabilities:
[tex]\[ \text{Probability of either blonde or black hair} = \frac{10}{26} + \frac{7}{26} = \frac{10 + 7}{26} = \frac{17}{26} \][/tex]
Thus, the probability that a student chosen from the class at random has either blonde hair or black hair is:
[tex]\[ \boxed{\frac{17}{26}} \][/tex]
