In a school, 80 students study one language only; either French or Italian. There are 20 more students studying French than Italian. 60% of those studying French are female. 30% of those studying Italian are male. A student is chosen at random from this group of students. (a) Complete the probability tree to show the probabilities associated with each branch. (b) What is the probability that the chosen student studies Italian and is a female? (c) What is the probability that the chosen student is female?

In a school, 80 students study one language only; either French or Italian. There are 20 more students studying French than Italian. 60% of those studying French are female. 30% of those studying Italian are male. A student is chosen at random from this group of students. (a) Complete the probability tree to show the probabilities associated with each branch. (b) What is the probability that the chosen student studies Italian and is a female? (c) What is the probability that the chosen student is female?

Answer

To solve the problem, we need to clarify the distribution of students studying French and Italian, and their gender breakdown based on given percentages. 1. **Finding number of students studying each language:** Let the number of students studying Italian be \( x \). Then, the number of students studying French would be \( x + 20 \). Since the total number of students is 80, we can set up the equation: \[ x + (x + 20) = 80 \] \[ 2x + 20 = 80 \] \[ 2x = 60 \] \[ x = 30 \] Thus, \( 30 \) students study Italian, and \( 50 \) students study French. 2. **Finding gender distribution:** - For French: - Total French students: 50 - 60% are female: \( 0.6 \times 50 = 30 \) females, - 40% are male: \( 0.4 \times 50 = 20 \) males. - For Italian: - Total Italian students: 30 - 30% are male: \( 0.3 \times 30 = 9 \) males, - 70% are female: \( 0.7 \times 30 = 21 \) females. 3. **Probability Tree Completion:** The probability tree would have the following branches: - Language Studied: - Italian - Male: \( P(M | I) = \frac{9}{80} \) - Female: \( P(F | I) = \frac{21}{80} \) - French - Male: \( P(M | F) = \frac{20}{80} \) - Female: \( P(F | F) = \frac{30}{80} \) 4. **Calculating Required Probabilities:** (b) **Probability that the chosen student studies Italian and is female:** \[ P(I \cap F) = P(I) \times P(F | I) = \frac{30}{80} \times \frac{21}{30} = \frac{21}{80} \] (c) **Probability that the chosen student is female:** \[ P(F) = P(F | I) \times P(I) + P(F | F) \times P(F) \] \[ P(F) = \frac{21}{80} + \frac{30}{80} = \frac{51}{80} \] Summarizing: (a) Complete the probability tree branches. (b) The probability that the chosen student studies Italian and is a female is \( \frac{21}{80} \) or 0.2625. (c) The probability that the chosen student is female is \( \frac{51}{80} \) or 0.6375.