Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 1 time and lost 9 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Goat
7. Car
8. Goat
9. Goat
10. Goat

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 321 times and lost 679 times.

1. Goat
2. Goat
3. Goat
4. Goat
5. Goat
6. Car
7. Car
8. Car
9. Car
10. Goat
11. Goat
12. Goat
13. Goat
14. Car
15. Goat
16. Goat
17. Goat
18. Goat
19. Goat
20. Goat
21. Goat
22. Goat
23. Car
24. Goat
25. Car
26. Car
27. Car
28. Goat
29. Goat
30. Goat
31. Goat
32. Goat
33. Goat
34. Car
35. Goat
36. Goat
37. Goat
38. Goat
39. Car
40. Goat
41. Goat
42. Car
43. Goat
44. Goat
45. Goat
46. Goat
47. Goat
48. Goat
49. Car
50. Goat
51. Goat
52. Goat
53. Goat
54. Goat
55. Goat
56. Car
57. Car
58. Car
59. Goat
60. Car
61. Goat
62. Goat
63. Car
64. Car
65. Car
66. Goat
67. Goat
68. Goat
69. Goat
70. Car
71. Goat
72. Car
73. Car
74. Goat
75. Goat
76. Goat
77. Goat
78. Goat
79. Goat
80. Goat
81. Goat
82. Goat
83. Goat
84. Goat
85. Car
86. Goat
87. Car
88. Goat
89. Car
90. Goat
91. Goat
92. Car
93. Goat
94. Goat
95. Goat
96. Goat
97. Goat
98. Car
99. Goat
100. Goat
101. Car
102. Car
103. Goat
104. Goat
105. Goat
106. Car
107. Car
108. Goat
109. Goat
110. Goat
111. Goat
112. Goat
113. Car
114. Goat
115. Goat
116. Goat
117. Car
118. Goat
119. Goat
120. Goat
121. Car
122. Goat
123. Goat
124. Goat
125. Goat
126. Goat
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Goat
133. Goat
134. Goat
135. Car
136. Car
137. Goat
138. Car
139. Car
140. Goat
141. Goat
142. Goat
143. Goat
144. Goat
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Car
151. Car
152. Goat
153. Goat
154. Goat
155. Goat
156. Goat
157. Car
158. Goat
159. Car
160. Goat
161. Goat
162. Goat
163. Goat
164. Goat
165. Goat
166. Goat
167. Goat
168. Goat
169. Goat
170. Goat
171. Car
172. Goat
173. Car
174. Goat
175. Car
176. Car
177. Goat
178. Car
179. Goat
180. Car
181. Goat
182. Goat
183. Goat
184. Goat
185. Goat
186. Car
187. Car
188. Goat
189. Goat
190. Goat
191. Goat
192. Goat
193. Car
194. Goat
195. Goat
196. Goat
197. Car
198. Car
199. Car
200. Goat
201. Car
202. Goat
203. Car
204. Goat
205. Car
206. Goat
207. Car
208. Car
209. Goat
210. Goat
211. Car
212. Goat
213. Goat
214. Goat
215. Goat
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Car
222. Goat
223. Goat
224. Goat
225. Goat
226. Goat
227. Goat
228. Goat
229. Goat
230. Goat
231. Car
232. Goat
233. Car
234. Goat
235. Car
236. Goat
237. Goat
238. Car
239. Car
240. Car
241. Goat
242. Goat
243. Goat
244. Goat
245. Car
246. Car
247. Car
248. Goat
249. Car
250. Goat
251. Goat
252. Goat
253. Goat
254. Goat
255. Goat
256. Car
257. Goat
258. Goat
259. Car
260. Goat
261. Car
262. Goat
263. Goat
264. Goat
265. Goat
266. Goat
267. Goat
268. Goat
269. Goat
270. Car
271. Goat
272. Goat
273. Car
274. Goat
275. Goat
276. Car
277. Goat
278. Goat
279. Goat
280. Car
281. Car
282. Goat
283. Car
284. Goat
285. Goat
286. Goat
287. Car
288. Goat
289. Goat
290. Goat
291. Goat
292. Goat
293. Car
294. Goat
295. Car
296. Goat
297. Car
298. Goat
299. Goat
300. Goat
301. Goat
302. Goat
303. Goat
304. Goat
305. Car
306. Car
307. Car
308. Goat
309. Goat
310. Car
311. Goat
312. Goat
313. Goat
314. Car
315. Goat
316. Goat
317. Goat
318. Goat
319. Goat
320. Car
321. Car
322. Car
323. Goat
324. Car
325. Goat
326. Goat
327. Car
328. Car
329. Car
330. Car
331. Goat
332. Goat
333. Goat
334. Goat
335. Car
336. Car
337. Car
338. Goat
339. Goat
340. Car
341. Goat
342. Goat
343. Goat
344. Goat
345. Goat
346. Goat
347. Car
348. Goat
349. Goat
350. Car
351. Goat
352. Car
353. Goat
354. Goat
355. Car
356. Goat
357. Goat
358. Goat
359. Goat
360. Car
361. Goat
362. Car
363. Goat
364. Car
365. Goat
366. Car
367. Goat
368. Goat
369. Goat
370. Car
371. Goat
372. Goat
373. Goat
374. Car
375. Goat
376. Goat
377. Goat
378. Goat
379. Goat
380. Goat
381. Goat
382. Goat
383. Goat
384. Goat
385. Goat
386. Goat
387. Car
388. Car
389. Goat
390. Car
391. Car
392. Goat
393. Car
394. Goat
395. Car
396. Goat
397. Car
398. Goat
399. Goat
400. Car
401. Goat
402. Goat
403. Car
404. Car
405. Goat
406. Car
407. Goat
408. Goat
409. Goat
410. Goat
411. Goat
412. Goat
413. Goat
414. Goat
415. Car
416. Goat
417. Car
418. Car
419. Goat
420. Goat
421. Car
422. Car
423. Goat
424. Goat
425. Goat
426. Goat
427. Goat
428. Goat
429. Goat
430. Goat
431. Goat
432. Car
433. Goat
434. Car
435. Car
436. Goat
437. Goat
438. Goat
439. Goat
440. Goat
441. Car
442. Goat
443. Goat
444. Goat
445. Car
446. Goat
447. Goat
448. Goat
449. Car
450. Goat
451. Car
452. Goat
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Car
460. Goat
461. Goat
462. Car
463. Goat
464. Car
465. Goat
466. Goat
467. Car
468. Car
469. Car
470. Goat
471. Goat
472. Goat
473. Goat
474. Car
475. Car
476. Car
477. Car
478. Car
479. Goat
480. Goat
481. Car
482. Goat
483. Car
484. Goat
485. Car
486. Car
487. Goat
488. Goat
489. Goat
490. Goat
491. Goat
492. Goat
493. Goat
494. Goat
495. Goat
496. Car
497. Car
498. Goat
499. Car
500. Goat
501. Goat
502. Goat
503. Goat
504. Car
505. Car
506. Goat
507. Goat
508. Goat
509. Goat
510. Goat
511. Goat
512. Goat
513. Goat
514. Goat
515. Car
516. Car
517. Goat
518. Goat
519. Car
520. Goat
521. Car
522. Goat
523. Car
524. Car
525. Car
526. Goat
527. Goat
528. Goat
529. Goat
530. Goat
531. Goat
532. Goat
533. Goat
534. Goat
535. Goat
536. Goat
537. Goat
538. Goat
539. Car
540. Goat
541. Goat
542. Goat
543. Car
544. Goat
545. Goat
546. Goat
547. Goat
548. Goat
549. Car
550. Car
551. Goat
552. Goat
553. Car
554. Goat
555. Goat
556. Car
557. Car
558. Goat
559. Goat
560. Car
561. Goat
562. Goat
563. Goat
564. Goat
565. Goat
566. Goat
567. Car
568. Goat
569. Car
570. Goat
571. Goat
572. Goat
573. Goat
574. Goat
575. Car
576. Goat
577. Goat
578. Car
579. Car
580. Car
581. Goat
582. Goat
583. Goat
584. Goat
585. Car
586. Goat
587. Goat
588. Goat
589. Goat
590. Car
591. Goat
592. Car
593. Car
594. Car
595. Goat
596. Goat
597. Car
598. Goat
599. Goat
600. Goat
601. Goat
602. Goat
603. Goat
604. Car
605. Goat
606. Goat
607. Goat
608. Goat
609. Goat
610. Car
611. Goat
612. Car
613. Goat
614. Car
615. Goat
616. Goat
617. Goat
618. Car
619. Car
620. Goat
621. Car
622. Goat
623. Car
624. Goat
625. Goat
626. Goat
627. Car
628. Car
629. Car
630. Car
631. Goat
632. Goat
633. Car
634. Car
635. Car
636. Goat
637. Car
638. Goat
639. Goat
640. Car
641. Goat
642. Car
643. Goat
644. Car
645. Goat
646. Car
647. Car
648. Car
649. Goat
650. Goat
651. Goat
652. Goat
653. Goat
654. Goat
655. Goat
656. Car
657. Car
658. Goat
659. Goat
660. Car
661. Goat
662. Goat
663. Car
664. Car
665. Car
666. Goat
667. Goat
668. Goat
669. Car
670. Goat
671. Goat
672. Goat
673. Car
674. Goat
675. Car
676. Car
677. Goat
678. Goat
679. Car
680. Car
681. Goat
682. Car
683. Goat
684. Goat
685. Goat
686. Goat
687. Goat
688. Goat
689. Goat
690. Goat
691. Goat
692. Goat
693. Goat
694. Goat
695. Car
696. Goat
697. Goat
698. Goat
699. Goat
700. Goat
701. Goat
702. Car
703. Car
704. Car
705. Goat
706. Goat
707. Goat
708. Goat
709. Car
710. Goat
711. Car
712. Car
713. Goat
714. Goat
715. Goat
716. Goat
717. Car
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Car
728. Car
729. Goat
730. Goat
731. Car
732. Goat
733. Goat
734. Car
735. Car
736. Goat
737. Goat
738. Goat
739. Goat
740. Goat
741. Goat
742. Goat
743. Car
744. Goat
745. Car
746. Goat
747. Goat
748. Goat
749. Goat
750. Car
751. Goat
752. Goat
753. Goat
754. Goat
755. Goat
756. Goat
757. Goat
758. Car
759. Goat
760. Goat
761. Goat
762. Car
763. Goat
764. Car
765. Goat
766. Car
767. Car
768. Car
769. Car
770. Car
771. Goat
772. Car
773. Goat
774. Car
775. Goat
776. Goat
777. Goat
778. Car
779. Goat
780. Goat
781. Goat
782. Goat
783. Goat
784. Car
785. Car
786. Goat
787. Goat
788. Car
789. Goat
790. Car
791. Goat
792. Goat
793. Goat
794. Goat
795. Goat
796. Goat
797. Goat
798. Car
799. Car
800. Car
801. Goat
802. Goat
803. Car
804. Car
805. Car
806. Car
807. Goat
808. Car
809. Car
810. Car
811. Goat
812. Goat
813. Goat
814. Goat
815. Goat
816. Goat
817. Goat
818. Car
819. Goat
820. Goat
821. Goat
822. Goat
823. Goat
824. Goat
825. Goat
826. Car
827. Goat
828. Goat
829. Car
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Car
836. Car
837. Goat
838. Goat
839. Car
840. Goat
841. Goat
842. Car
843. Goat
844. Car
845. Car
846. Goat
847. Car
848. Goat
849. Goat
850. Car
851. Car
852. Goat
853. Goat
854. Goat
855. Car
856. Goat
857. Goat
858. Goat
859. Car
860. Car
861. Goat
862. Goat
863. Goat
864. Car
865. Goat
866. Car
867. Goat
868. Car
869. Goat
870. Car
871. Car
872. Goat
873. Car
874. Car
875. Goat
876. Goat
877. Car
878. Goat
879. Goat
880. Car
881. Goat
882. Goat
883. Goat
884. Goat
885. Goat
886. Goat
887. Goat
888. Goat
889. Car
890. Car
891. Goat
892. Goat
893. Car
894. Car
895. Car
896. Goat
897. Goat
898. Goat
899. Goat
900. Goat
901. Car
902. Goat
903. Goat
904. Goat
905. Goat
906. Goat
907. Car
908. Goat
909. Goat
910. Goat
911. Car
912. Car
913. Goat
914. Goat
915. Goat
916. Goat
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Car
923. Goat
924. Goat
925. Goat
926. Car
927. Goat
928. Car
929. Car
930. Goat
931. Goat
932. Car
933. Goat
934. Goat
935. Car
936. Goat
937. Goat
938. Goat
939. Car
940. Goat
941. Goat
942. Goat
943. Car
944. Goat
945. Goat
946. Goat
947. Car
948. Car
949. Goat
950. Car
951. Car
952. Goat
953. Car
954. Goat
955. Goat
956. Goat
957. Car
958. Goat
959. Goat
960. Goat
961. Goat
962. Car
963. Goat
964. Goat
965. Goat
966. Car
967. Goat
968. Goat
969. Goat
970. Goat
971. Goat
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Goat
979. Car
980. Car
981. Goat
982. Goat
983. Goat
984. Goat
985. Car
986. Car
987. Car
988. Car
989. Car
990. Goat
991. Car
992. Car
993. Goat
994. Car
995. Goat
996. Car
997. Car
998. Car
999. Goat
1000. Goat

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 109 times and lost 891 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 30 times and lost 70 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}