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 4 times and lost 6 times.

1. Goat
2. Car
3. Goat
4. Car
5. Goat
6. Goat
7. Car
8. Goat
9. Car
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 315 times and lost 685 times.

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