Porisma

Worked solution

Model red cars and solve a zero-or-one probability

A fixed number n of cars is observed and R counts those that are red. (i) State in context two assumptions for a binomial model. Assume now R∼B(n,p)R\sim B(n,p). (ii) For n=20 and p=0.15 find P(4≤R<8)P(4\le R<8). (iii) For n=20 and P(R=0 or 1)=0.2P(R=0\text{ or }1)=0.2, write an equation for p and find p numerically.

[2009/PII/11(part)]

CJC · Lecture · Example 8

(i) State the car assumptions

The probability that any one car is red stays the same throughout the sample, and the colour of each car is independent of the colours of the other cars. These are the two Teacher completions.

How did this part go?

(ii) Convert the mixed interval

For R∼B(20,0.15)R\sim B(20,0.15), P(4≤R<8)=P(R≤7)−P(R≤3)=0.346P(4\le R<8)=P(R\le7)-P(R\le3)=0.346 to 3 significant figures. The upper bound 8 is excluded; the lower bound 4 is included.

How did this part go?

(iii) Write the equation

For R∼B(20,p)R\sim B(20,p), P(R=0 or 1)=P(R=0)+P(R=1)=0.2P(R=0\text{ or }1)=P(R=0)+P(R=1)=0.2, so (1−p)20+20p(1−p)19=0.2(1-p)^{20}+20p(1-p)^{19}=0.2. This retains the complete Teacher working.

How did this part go?

(iii) Solve with the source graph procedure

Enter the left side as Y1 and 0.2 as Y2. Adjust the window, then find the intersection; its x-coordinate is p. The source window is Xmin 0, Xmax 1, Xscl 0, Ymin 0, Ymax 0.5, Yscl 1, Xres 1 (derived ΔX≈0.003787879, TraceStep≈0.007575758). The source plot gives p≈0.1424323p\approx0.1424323, hence 0.142 to 3 significant figures.

How did this part go?

Compare your sketch

Solve for the probability that a car is red

  1. 1. Write the probability equation

    Zero or one red car in 20 has probability 0.2. Add these disjoint masses to write (1−p)²⁰+20p(1−p)¹⁹=0.2.

  2. 2. Enter both sides in the Y-editor

    The calculator X now stands for the unknown probability p.

    Y1
    (1-X)^20+20X(1-X)^19
    Y2
    0.2

  3. 3. Choose a window that shows the crossing

    The retained source uses these bounds. Its complete settings and plotted screens remain below.

    Xmin
    0
    Xmax
    1
    Ymin
    0
    Ymax
    0.5

  4. 4. Calculate the intersection

    Press 2nd → TRACE, choose intersect, select the two curves and provide a nearby guess. Read the x-coordinate as p, not the y-coordinate.

    p ≈ 0.142432347; to 3 significant figures: 0.142

Check 0 ≤ p ≤ 1. This unknown is continuous: an integer-step sample-size table is not the required numerical root solve. If asked for an equation, write it before using the calculator. Displayed intersection values are approximate.

View original calculator screens
Expand diagram

Use the scrollbars or arrow keys to inspect details.

Calculator setup, full source window and analytical reconstruction of the intersection plot. (1-X)^20+20X(1-X)^19, Y2=0.2; X range 0 to 1, Y range 0 to 0.5; intersection p≈0.142432347. The formula is plotted directly, rather than embedding the raster screenshot.
Calculator setup, full source window and analytical reconstruction of the intersection plot. (1-X)^20+20X(1-X)^19, Y2=0.2; X range 0 to 1, Y range 0 to 0.5; intersection p≈0.142432347. The formula is plotted directly, rather than embedding the raster screenshot.
Calculator setup, full source window and analytical reconstruction of the intersection plot. (1-X)^20+20X(1-X)^19, Y2=0.2; X range 0 to 1, Y range 0 to 0.5; intersection p≈0.142432347. The formula is plotted directly, rather than embedding the raster screenshot.
Open original screens (new tab)

Common mistakes

For “zero or one”, add the two disjoint masses before solving for p.

Read the hint