Binomial Distribution worksheet

1. P(X = 7): 0.16

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(9, 7) = 36 ways to choose which 7 trials succeed

   $C\left(9,7\right)$$=36$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 7)}$$=36·{0.6}^7·{0.4}^2$$=0.161243136$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 7)}$$=0.929456128$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 7)}=0.768212992,\text{P(X > 7)}=0.070543872,\text{P(X ≥ 7)}=0.231787008$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =9\times 0.6=5.4,\sigma =\sqrt{9\times 0.6\times 0.4}=1.4696938457$

2. P(X = 4): 0.23

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(8, 4) = 70 ways to choose which 4 trials succeed

   $C\left(8,4\right)$$=70$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 4)}$$=70·{0.4}^4·{0.6}^4$$=0.2322432$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 4)}$$=0.8263296$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 4)}=0.5940864,\text{P(X > 4)}=0.1736704,\text{P(X ≥ 4)}=0.4059136$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =8\times 0.4=3.2,\sigma =\sqrt{8\times 0.4\times 0.6}=1.3856406461$

3. P(X = 7): 0.24

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(11, 7) = 330 ways to choose which 7 trials succeed

   $C\left(11,7\right)$$=330$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 7)}$$=330·{0.6}^7·{0.4}^4$$=0.2364899328$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 7)}$$=0.7037157376$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 7)}=0.4672258048,\text{P(X > 7)}=0.2962842624,\text{P(X ≥ 7)}=0.5327741952$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =11\times 0.6=6.6,\sigma =\sqrt{11\times 0.6\times 0.4}=1.6248076809$

4. P(X = 5): 0.17

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(9, 5) = 126 ways to choose which 5 trials succeed

   $C\left(9,5\right)$$=126$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 5)}$$=126·{0.4}^5·{0.6}^4$$=0.167215104$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 5)}$$=0.900647424$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 5)}=0.73343232,\text{P(X > 5)}=0.099352576,\text{P(X ≥ 5)}=0.26656768$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =9\times 0.4=3.6,\sigma =\sqrt{9\times 0.4\times 0.6}=1.4696938457$

5. P(X = 9): 0.03

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(11, 9) = 55 ways to choose which 9 trials succeed

   $C\left(11,9\right)$$=55$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 9)}$$=55·{0.5}^9·{0.5}^2$$=0.02685546875$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 9)}$$=0.994140625$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 9)}=0.96728515625,\text{P(X > 9)}=0.005859375,\text{P(X ≥ 9)}=0.03271484375$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =11\times 0.5=5.5,\sigma =\sqrt{11\times 0.5\times 0.5}=1.6583123952$

6. P(X = 7): 0.23

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(12, 7) = 792 ways to choose which 7 trials succeed

   $C\left(12,7\right)$$=792$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 7)}$$=792·{0.6}^7·{0.4}^5$$=0.227030335488$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 7)}$$=0.56182177792$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 7)}=0.334791442432,\text{P(X > 7)}=0.43817822208,\text{P(X ≥ 7)}=0.665208557568$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =12\times 0.6=7.2,\sigma =\sqrt{12\times 0.6\times 0.4}=1.6970562748$

7. P(X = 6): 0.18

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(12, 6) = 924 ways to choose which 6 trials succeed

   $C\left(12,6\right)$$=924$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 6)}$$=924·{0.4}^6·{0.6}^6$$=0.176579149824$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 6)}$$=0.841787707392$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 6)}=0.665208557568,\text{P(X > 6)}=0.158212292608,\text{P(X ≥ 6)}=0.334791442432$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =12\times 0.4=4.8,\sigma =\sqrt{12\times 0.4\times 0.6}=1.6970562748$

8. P(X = 6): 0.23

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(12, 6) = 924 ways to choose which 6 trials succeed

   $C\left(12,6\right)$$=924$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 6)}$$=924·{0.5}^6·{0.5}^6$$=0.2255859375$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 6)}$$=0.61279296875$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 6)}=0.38720703125,\text{P(X > 6)}=0.38720703125,\text{P(X ≥ 6)}=0.61279296875$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =12\times 0.5=6,\sigma =\sqrt{12\times 0.5\times 0.5}=1.7320508076$

9. P(X = 6): 0.05

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(7, 6) = 7 ways to choose which 6 trials succeed

   $C\left(7,6\right)$$=7$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 6)}$$=7·{0.5}^6·{0.5}^1$$=0.0546875$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 6)}$$=0.9921875$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 6)}=0.9375,\text{P(X > 6)}=0.0078125,\text{P(X ≥ 6)}=0.0625$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =7\times 0.5=3.5,\sigma =\sqrt{7\times 0.5\times 0.5}=1.3228756555$

10. P(X = 5): 0.23

1. The binomial formula counts the ways to place k successes among n trials. Then it multiplies by the chance of each of those orders

   $\text{P(X = k)}$$=C\left(n,k\right)$$=·$$=p^k$$=·$$={\left(1-p\right)}^{n-k}$

2. There are C(11, 5) = 462 ways to choose which 5 trials succeed

   $C\left(11,5\right)$$=462$

3. Put in your numbers to get the chance of exactly k successes

   $\text{P(X = 5)}$$=462·{0.5}^5·{0.5}^6$$=0.2255859375$

4. Add up the chances of 0 through k successes to get the chance of k or fewer. (The calculator uses a special function, the regularized incomplete beta, that gives the same total quickly.)

   $\text{P(X ≤ 5)}$$=0.5$

5. All the chances add up to 1, so the others come from subtracting. P(X < k) = P(X ≤ k) − P(X = k). P(X > k) = 1 − P(X ≤ k). P(X ≥ k) = 1 − P(X < k)

   $\text{P(X < 5)}=0.2744140625,\text{P(X > 5)}=0.5,\text{P(X ≥ 5)}=0.7255859375$

6. The average number of successes (mean) is n·p. The variance is n·p·(1 − p), and its square root, the standard deviation, shows the typical spread

   $\mu =11\times 0.5=5.5,\sigma =\sqrt{11\times 0.5\times 0.5}=1.6583123952$