Probability and Statistics: Making Sense of Randomness
What probability means, how the bell curve works, what z-scores and confidence intervals actually tell you, and why sample size matters.
Your electricity bill charges you for kilowatt-hours (kWh), but appliances are rated in watts (W). Understanding the relationship between these two units is the key to knowing what actually costs money on your electric bill — and what does not.
A watt measures how fast energy is being used right now. It is a rate, like miles per hour for a car. A 100W light bulb uses energy at a rate of 100 watts — but only while it is turned on.
Named after James Watt (the steam engine inventor), one watt equals one joule of energy per second. A 60W bulb converts 60 joules every second into light and heat.
A kilowatt-hour (kWh) measures the total energy consumed over time. It is a quantity, like miles driven. If a 1000W (1 kW) appliance runs for 1 hour, it uses 1 kWh.
The formula: kWh = watts x hours / 1000
100 x 10 / 1000 = 1 kWh1500 x 8 / 1000 = 12 kWh10 x 24 / 1000 = 0.24 kWhYour utility company charges a rate per kWh. The US average is roughly $0.12/kWh, but it ranges from about $0.08 (Louisiana) to $0.30+ (Hawaii, Connecticut).
Cost formula: cost = kWh x rate per kWh
watts x hours / 1000kWh x $/kWhCost depends on both wattage and runtime. A 1500W space heater running 8 hours/day costs ~$1.44/day ($43/month). A 150W refrigerator running 24/7 costs ~$0.43/day ($13/month). The heater uses 10x the power but the fridge runs 3x longer — yet the heater still costs 3x more because the wattage difference dominates.
Many devices draw power even when "off" — TVs in standby, phone chargers plugged in with no phone, game consoles, routers. This phantom load typically accounts for 5-10% of a household's electricity bill.
A single device might only draw 1-5W in standby, which seems trivial. But multiply by 20+ devices running 24/7/365 and it adds up: 50W of phantom load = 438 kWh/year = ~$53/year.
Your electric meter measures total kWh consumed. The utility reads it monthly and charges you for the difference since the last reading. Smart meters transmit readings automatically and can show you real-time consumption.
What probability means, how the bell curve works, what z-scores and confidence intervals actually tell you, and why sample size matters.
Area and volume formulas for every common shape, the Pythagorean theorem, Law of Sines and Cosines, and slope of a line.
How fractions work, why prime factorisation matters, the GCF and LCM connection, ratios, proportions, and percentages as fractions of 100.