This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the Mathematics questions. SUBJECT: MATHEMATICS 1. A coin tossed three times, find the probability of getting: Step 1: Determine the sample space. When a coin is tossed three times, the total number of possible outcomes is 2^3 = 8. The sample space S is: S = \HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\ (i) 3 heads The event of getting 3 heads is E_1 = \HHH\. The number of outcomes in E_1 is 1. P(3 heads) = Number of outcomes in E_1Total number of outcomes = (1)/(8) The probability of getting 3 heads is (1)/(8). (ii) Not 3 heads This is the complement of getting 3 heads. P(Not 3 heads) = 1 - P(3 heads) = 1 - (1)/(8) = (7)/(8) The probability of not getting 3 heads is (7)/(8). (iii) At least 2 heads The event of getting at least 2 heads means getting 2 heads or 3 heads. Outcomes with 2 heads: \HHT, HTH, THH\ Outcomes with 3 heads: \HHH\ The event E_3 = \HHT, HTH, THH, HHH\. The number of outcomes in E_3 is 4. P(At least 2 heads) = Number of outcomes in E_3Total number of outcomes = (4)/(8) = (1)/(2) The probability of getting at least 2 heads is (1)/(2). (iv) At least two tails The event of getting at least two tails means getting 2 tails or 3 tails. Outcomes with 2 tails: \HTT, THT, TTH\ Outcomes with 3 tails: \TTT\ The event E_4 = \HTT, THT, TTH, TTT\. The number of outcomes in E_4 is 4. P(At least two tails) = Number of outcomes in E_4Total number of outcomes = (4)/(8) = (1)/(2) The probability of getting at least two tails is (1)/(2). 2. The favorite fruits of 40 students are in the table below: | Fruit | Oranges | Apples | Mangoes | Grapes | | :-------- | :------ | :----- | :------ | :----- | | Frequency | 15 | 7 | 8 | 10 | Step 1: Calculate the angles for the pie chart. The total number of students is 40. The total angle in a circle is 360^. Angle for Oranges: (15)/(40) × 360^ = 135^ Angle for Apples: (7)/(40) × 360^ = 63^ Angle for Mangoes: (8)/(40) × 360^ = 72^ Angle for Grapes: (10)/(40) × 360^ = 90^ Note: I cannot draw the pie chart in this format, but these are the calculations needed to construct it. Step 2: Calculate the fraction of mangoes. The frequency of mangoes is 8. The total number of students is 40. Fraction of mangoes = Frequency of MangoesTotal number of students = (8)/(40) = (1)/(5) The fraction of mangoes is (1)/(5). Send me the next one 📸