Calculating [latex]P(X \ge x)[/latex] is similar to calculating [latex]P(X \gt  x)[/latex] with only one difference:

• [latex]P(X \ge x)[/latex], the probability of at least [latex]x[/latex] successes, includes the [latex]x[/latex] value.
• [latex]P(X \gt  x)[/latex], the probability of more than [latex]x[/latex] successes, does not include the [latex]x[/latex] value.

In other words:

• [latex]P(X \gt x) = 1- P(X \le x) = 1-\text{BINOM.DIST}(x, n, p, 1)[/latex]
• [latex]P(X \ge x) = 1- P(X \le x-1) = 1-\text{BINOM.DIST}(x-1, n, p, 1)[/latex]

Why do we use [latex]x-1[/latex] in the above formula?

• It has to do with what it means to take a complement.
• When taking a complement, we take all values outside of that sample space.
• Since [latex]x[/latex] is included in the range, we ‘stop’ at [latex]x-1[/latex] when taking the complement: [latex]P(X \ge x) = 1 - [P(X=0)+P(X=1)+...+P(X=x-1)][/latex]

Using Formulas to Calculate At least (Example)

Let us revisit the hotel example from the previous section to highlight the similarities and differences between ‘more than’ and ‘at least’ calculations.

Example 28.1 – Using the Formula

Problem Setup: A hotel’s records indicate that 65% of its guests are visitors from Canada.

Question: From a random sample of 12 guests, what is the probability that at least 10 of them are from Canada?

Solution: We know the following:

• [latex]P(\text{at least 10}) = P(x\ge 10) = P(x=10)+P(x=11) + P(x=12)[/latex]
• [latex]n=12[/latex] and [latex]p=0.65[/latex].

This gives:

• [latex]P(x=10) = {}_{12}C_{10} \cdot (0.65)^{10} \cdot (1 - 0.65)^{12-10} = \frac{12!}{10!2!} \cdot (0.65)^{10} \cdot (0.35)^2 =66(0.01346)(0.1225) = 0.10885[/latex]
• [latex]P(x=11) = {}_{12}C_{11} \cdot (0.65)^{11} \cdot (1 - 0.65)^{12-11} = \frac{12!}{11!1!} \cdot (0.65)^{11} \cdot (0.35)^1 =12(0.00875)(0.35) = 0.03675[/latex]
• [latex]P(x=12) = {}_{12}C_{12} \cdot (0.65)^{12} \cdot (1 - 0.65)^{12-12} = \frac{12!}{12!0!} \cdot (0.65)^{12} \cdot (0.35)^0 =1(0.00569)(1) = 0.00569[/latex]
• [latex]P(x\ge 10) =  P(x=10)+P(x=11)+P(x=12) = 0.10885+0.03675+ 0.00569 = 0.1513[/latex]

Conclusion: There is an 15.13% chance that at least 10 of them are from Canada.

Using Excel to Calculate At least (VIDEO)

• Using Excel’s BINOM.DIST() function is much quicker than using the formula shown in the previous section
• Again, just be careful of which [latex]x[/latex] value to include and don’t forget to take the complement.
• We will try this out in the next example.

Example 28.2 – Using Excel and a Complement

Problem Setup: Let us now revisit example 28.1 but we will use Excel.

Question: Can you use Excel’s BINOM.DIST to calculate the probability of at least 10 guests being from Canada?

Solution: Click here to download the Excel solution file. Also, see the video below:

Conclusion: Again, here is an 15.13% chance that at least 10 of them are from Canada.

Key Takeaways (EXERCISE)

Your Own Notes (EXERCISE)

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