Example: Tossing a Biased Coin Suppose you have a biased coin that lands on heads $70$% of the time. You flip the coin $100$ times. What is the probability of getting between $65$ and $75$ heads (inclusive)?

Step 1: Define the Binomial Distribution

• Number of trials (n): $100$
• Probability of success (p): $0.7$
• Random variable X: Number of heads in $100$ $flips$.

Step 2: Check Conditions for Normal Approximation

• $np = 100 \times 0.7 = 70$
• $n(1-p) = 100 \times 0.3 = 30$ Both $np$ and $n(1-p)$ are greater than $5$, so the Normal approximation is appropriate.

Step 3: Determine the Mean (μ) and Standard Deviation (σ)

• Mean (μ): $np = 70$
• Standard deviation (σ):

Step 4: Apply the Continuity Correction Since the Normal distribution is continuous and the Binomial is discrete, apply a continuity correction when converting Binomial probabilities to Normal probabilities. This involves adjusting the bounds by $0.5$.

• To find $P(65 \leq X \leq 75)$ in the Binomial distribution, approximate this by $P(64.5 < X < 75.5)$ in the Normal distribution.

Step 5: Convert to Z-scores Convert the adjusted values to $z-scores$:

Step 6: Use the Z-table Look up the $z-scores$ in the $z-table$ (or use a calculator):

• $P(Z < -1.20) \approx 0.1151$
• $P(Z < 1.20) \approx 0.8849$

Step 7: Calculate the Required Probability Subtract the smaller probability from the larger one to find the probability of getting between $65$ and $75$ $heads$:

So, the probability of getting between $65$ and $75$ $heads$ is approximately 0.770.