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question:Which of the following statements is incorrect? ( ) A: Frequency is random and cannot be determined before the experiment. As the number of experiments increases, the frequency generally tends to approach the probability. B: If the probability of a random event A occurring is P(A), then 0 leq P(A) leq 1. C: Mutually exclusive events are not necessarily complementary events, but complementary events are definitely mutually exclusive events. D: Among 5 lottery tickets, one is a winning ticket. If person A draws first and person B draws later, then the possibility of B drawing the winning ticket is the same as that of A.
answer:The frequency stabilizes at a certain constant, which is called probability. Therefore, option A is incorrect, while options B, C, and D are correct. Thus, the incorrect statement is boxed{text{A}}.
question:James is sitting outside, counting how many people go past him in vehicles. He saw 12 trucks, a couple of buses, twice as many taxis, some motorbikes, and 30 cars. If the trucks held 2 people each, the buses held 15 people each, the taxis held 2 people each, the motorbikes held 1 person each, and the cars held 3 people each, James has seen 156 passengers today. How many vehicles has James counted so far today?
answer:Let's calculate the number of passengers in each type of vehicle: - Trucks: 12 trucks * 2 people each = 24 people - Buses: A couple of buses means 2 buses * 15 people each = 30 people - Taxis: Twice as many taxis as buses means 2 * 2 = 4 taxis * 2 people each = 8 people - Cars: 30 cars * 3 people each = 90 people We are given that James has seen 156 passengers today. We have the number of passengers for trucks, buses, taxis, and cars, but we don't know the number of motorbikes yet. Let's subtract the known passengers from the total to find out how many passengers were on motorbikes: 156 total passengers - 24 (trucks) - 30 (buses) - 8 (taxis) - 90 (cars) = 4 passengers on motorbikes Since each motorbike held 1 person, there were 4 motorbikes. Now let's add up all the vehicles: - Trucks: 12 - Buses: 2 - Taxis: 4 - Motorbikes: 4 - Cars: 30 Total vehicles = 12 + 2 + 4 + 4 + 30 = 52 vehicles James has counted boxed{52} vehicles so far today.
question:f 2 is a root of the equation x^{2}3x+k=0, then the value of the constant k is ( ) A: 1 B: 2 C: 1 D: 2
answer:Given that 2 is a root of the equation x^{2}-3x+k=0, we substitute x=2 into the equation to find the value of k. Starting with the equation: [x^{2}-3x+k=0] Substituting x=2: [2^{2}-3times 2+k=0] Simplifying the equation step by step: [4-6+k=0] [k=2] Therefore, the value of the constant k is 2. According to the given options: A: 1 B: 2 C: -1 D: -2 The correct answer is boxed{B}.
question:30 people attend a gathering. Each person shakes hands with at least three other people. What is the minimum possible number of handshakes?
answer:We need to find the minimum number of handshakes if each person shakes hands with exactly three other people. For each handshake involving two people, if everyone shakes hands with three others, this leads to a total of 30 times 3 = 90 handshakes. However, each handshake involves two people, so the total must be divided by 2 to avoid double-counting: [ frac{30 times 3}{2} = boxed{45} ] To achieve 45 handshakes, we could arrange the participants in a circle, and each person shakes hands with the person to their immediate left, immediate right, and one additional person. For simplicity, the additional person could be two seats away clockwise. This ensures each person is involved in exactly three handshakes.