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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I can help you with 4.2, 4.3, and 4.4. I don't have question 4.5 in the images you sent, so please send it if you'd like me to solve it! Here are the solutions for the questions you asked about: 4.2 The probability that a randomly selected community hub has cameras and reported vandalism is calculated as follows: Step 1: Identify the relevant probabilities from the data or the tree diagram. P(Cameras Installed) = P(C) = (100)/(200) = (1)/(2) P(Vandalism | Cameras Installed) = P(V|C) = (10)/(100) = (1)/(10) Step 2: Multiply these probabilities to find the joint probability. P(C V) = P(C) × P(V|C) = (1)/(2) × (1)/(10) = (1)/(20) The probability is (1)/(20). 4.3 The probability that a community hub did not have cameras and experienced vandalism, as a percentage, is calculated as follows: Step 1: Identify the relevant probabilities from the data or the tree diagram. P(No Cameras Installed) = P(NC) = (100)/(200) = (1)/(2) P(Vandalism | No Cameras Installed) = P(V|NC) = (15)/(100) = (3)/(20) Step 2: Multiply these probabilities to find the joint probability. P(NC V) = P(NC) × P(V|NC) = (1)/(2) × (3)/(20) = (3)/(40) Step 3: Convert the probability to a percentage. (3)/(40) × 100\% = 0.075 × 100\% = 7.5\% The probability is 7.5\%. 4.4 To determine which group of community hubs has a higher risk of vandalism, we compare the conditional probabilities of vandalism for each group: Step 1: State the conditional probability of vandalism given cameras installed. P(V|C) = (10)/(100) = (1)/(10) Step 2: State the conditional probability of vandalism given no cameras installed. P(V|NC) = (15)/(100) = (3)/(20) Step 3: Compare the two probabilities. To compare (1)/(10) and (3)/(20), we can express (1)/(10) as (2)/(20). Since (3)/(20) > (2)/(20), the probability of vandalism is higher when no cameras are installed. The group of community hubs with a higher risk of vandalism is hubs with no cameras installed. This is because the probability of vandalism given no cameras (P(V|NC) = (3)/(20)) is greater than the probability of vandalism given cameras (P(V|C) = (1)/(10) = (2)/(20)). Send me the next one 📸